Astron.  Of  r>*. 


NAVIGATION 


THE  MACMILLAN  COMPANY 

NEW  YORK    •    BOSTON   •    CHICAGO    •    DALLAS 
ATLANTA    •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON    •    BOMBAY   •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


NAVIGATION 


BY 

HAROLD   JACOBY 

RUTHERFURD   PROFESSOR    OF   ASTRONOMY 
IN    COLUMBIA    UNIVERSITY 


Nefo  gorfc 

THE   MACMILLAN   COMPANY 
1917 

All  rights  reserved 


V    "V 

*« vent.  D»»^t. 


X 


COPTKIGHT,   1917, 

BY  THE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  October,  1917. 


Xorfcoooti  Press 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


MACLEAB  JACOBY 

QUARTERMASTER,    THIRD    CLASS,    U.  8.  N. 

ENLISTED    FOR    THE    PERIOD    OF    THE    WAR 

THIS    VOLUME    IS    OFFERED   AS 

A    MARK    OF    RESPECT 

BY   HIS    FATHER 


3C754o 


PREFACE 

THE  present  volume  was  undertaken  with  certain  very 
definite  aims.  In  the  first  place,  it  is  intended  to  be  com- 
plete in  itself,  so  that  it  should  be  possible  to  navigate  a  ship 
in  any  ocean  not  very  near  the  north  or  south  pole  without 
other  books  or  tabular  works,  excepting  only  the  nautical 
almanac  for  the  year  in  which  the  voyage  is  made.  To  attain 
this  end  without  unduly  extending  the  size  of  the  volume, 
certain  essential  nautical  tables  have  been  abridged ;  but 
all  are  given  in  sufficiently  extended  form  to  permit  of  actual 
navigation  with  their  aid;  and  they  are  especially  suitable 
for  beginners,  who  can  here  attain  the  necessary  knowledge 
with  less  effort  than  would  be  necessary  with  more  bulky 
volumes.  In  cases  where  very  extended  tables  are  conven- 
ient, they  are  mentioned  in  the  text. 

In  the  second  place,  the  author  has  not  assumed  that  the 
reader  possesses  formal  mathematical  and  astronomical 
knowledge,  or  desires  to  possess  such  knowledge.  When- 
ever methods  of  navigation  require  for  their  demonstration 
an  understanding  of  spherical  trigonometry,  or  some  other 
branch  of  formal  mathematical  science,  such  demonstrations 
have  been  replaced  with  incomplete  or  ''outline  "  demonstra- 
tions designed  for  the  non-mathematical  reader.  Practical 
methods  are  fully  explained ;  and  an  attempt  has  always 
been  made  so  to  word  the  explanations  that  the  reader, 
even  the  beginner,  will  understand  his  problem,  and  will 
know  what  he  is  doing,  and  why  he  does  it. 

The  requirements  of  those  who  may  study  without  a 
teacher  have  received  constant  and  special  attention.  To 
meet  these  requirements  the  whole  subject  is  presented  in 

vii 


viii  PREFACE 

a  somewhat  informal  manner;  such  topics  as  the  use  of 
logarithms,  or  the  principles  on  which  all  mathematical 
tables  are  constructed  —  these  less  attractive  parts  of  the 
subject  are  not  presented  in  a  special  chapter,  but  are  de- 
scribed in  a  sort  of  digression,  when  needed  in  the  discussion 
of  an  actual  navigational  problem. 

Finally,  to  further  simplify  and  condense  his  material, 
the  author  has  made  no  attempt  to  include  every  method 
that  can  possibly  be  used  to  navigate  a  ship,  or  that  ever  has 
been  used  to  navigate  a  ship ;  his  purpose  has  been  rather 
to  limit  the  volume  to  the  methods  at  present  thought  best 
by  the  most  reliable  modern  authorities. 

Other  books  on  navigation  have  been  used  freely,  espe- 
cially in  the  preparation  of  the  tables.  Among  these,  that 
admirable  encyclopedia  of  navigation,  known  as  "Bowditch," 
published  by  the  Hydrographic  Office,  United  States  Navy, 
and  Kelvin's  " Tables  for  Sumner's  Method  at  Sea"  have 
been  found  of  the  greatest  help. 

Miss  Dorothy  W.  Block,  Instructor  of  Astronomy  in 
Hunter  College,  New  York,  has  helped  with  great  energy 
in  the  preparation  of  the  tables  and  the  correction  of  the 
text.  It  is  hoped  that  such  errors  as  may  now  remain  in 
the  book  are  few  in  number. 

H.  J. 

COLUMBIA  UNIVERSITY, 
August,  1917. 


TABLE   OF   CONTENTS 

CHAPTER         .  PAGE 

I.    THE  FUNDAMENTAL  PROBLEM  OF  NAVIGATION   .         .         1 

The  problem  stated.  Reasons  for  the  existence  of  the 
problem.  Definition  of  "ship's  position."  Longitude 
meridians  and  latitude  parallels.  Greenwich  the  initial 
meridian.  Position  determined  by  observation;  on  the 
coast  and  at  sea.  Dead  reckoning.  Sextant  observa- 
tions. Chronometer. 

II.    DEAD  RECKONING  WITHOUT  LOGARITHMS     ...         7 

The  two  problems.  Designation  of  .ship's  course. 
Latitude  difference  and  departure.  The  traverse  table. 
Use  and  construction  of  tables  in  general.  Arguments 
and  tabular  numbers.  Relation  between  departure  and 
longitude  difference.  Middle  latitude. 

III.  DEAD  RECKONING  WITH  LOGARITHMS  ...       23 
Explanation    of    number    logarithms    and    their   use. 

Multiplication  and  division.  Trigonometric  logarithms. 
Solution  of  the  two  problems.  Middle  latitude  sailing. 
Mercator  sailing.  Meridional  parts.  Great  circle  sail- 
ing. The  rhumb  line.  Composite  sailing.  Parallel 
sailing.  Traverse  sailing. 

IV.  THE  COMPASS 40 

The  card,  how  divided.     Degrees  and  points.     Boxing 

the  compass.  Lubber  line.  True  course  and  compass 
course.  Error,  variation  and  deviation.  Swinging  ship. 
Azimuth  circle  and  pelorus.  The  compass  formulas. 
The  two  deviation  tables.  Comparative  table  of  points 
and  degrees. 

V.   COASTWISE  NAVIGATION 53 

The  "fix."  Bow  bearings.  Doubling  the  bearing  on 
the  bow.  Bow  and  beam  bearings.  Distance  a-beam. 
Cross  bearings.  The  danger  angle.  Danger  bearing. 
Soundings. 

ix 


X  TABLE   OF  CONTENTS 

CHAPTER  PAGE 

VI.    THE  SEXTANT       .  .61 

Description  of  the  instrument  and  its  use.  The  vernier. 
Index  error.  Three  adjustments.  The  artificial  horizon. 
Correcting  the  altitude.  Dip.  Refraction.  Parallax. 

VII.   THE  NAUTICAL  ALMANAC     .         .         .         .  .75 

Specimen  pages  of  it.  Greenwich  mean  time.  Decli- 
nation. Equation  of  time.  Astronomic  and  civil  day. 
Apparent  solar  time.  Chronometers  and  the  rate  card. 
Right  ascension.  Solar  and  sidereal  time. 

VIII.    OLDER  NAVIGATION  METHODS     .....       86 

The  noon-sight  for  latitude.  Tropic  observations  and 
the  midnight  sun  in  high  latitudes.  Preparing  for  the 
observation.  Setting  the  cabin  clock.  Star  observa- 
tion. Ex-meridian  observation.  The  time-sight  for 
longitude.  Set  of  current.  Star  time-sight.  Condensed 
forms  of  calculation. 

IX.   NEWER  NAVIGATION  METHODS    .         .         .         .         .     108 

Errors  produced  by  dead  reckoning.  Captain  Sumner, 
and  the  Sumner  line.  Bearing  of  the  line.  The  Sumner 
point.  Azimuth  tables.  Condensed  form  of  calcula- 
tion. Star  observations.  Comparison  of  Sumner  navi- 
gation with  time-sight  navigation.  The  Kelvin  table. 
Condensed  forms  of  sun  and  star  observations.  Inter- 
section of  two  Sumner  lines  obtained  with  a  special  table. 
Motion  of  ship  between  observations. 

X.   A  NAVIGATOR'S  DAY  AT  SEA       .         .         .         .         .141 

Voyage  planned  from  New  York  to  Colon.  Departure 
at  Sandy  Hook  lightship.  The  course  to  Watlings  Island. 
The  variation  and  deviation  applied.  Azimuth  of  the  sun 
observed  at  sunrise.  Bow  and  beam  bearings  of  Barnegat 
Light.  The  patent  log  and  the  log  book.  New  course 
from  Barnegat.  Morning  sight  worked  as  a  Sumner  line. 
Another  Azimuth  observation.  Weather  thickens  at 
11 :  30.  Ex-meridian  sight  at  11  :  42,  worked  as  a  Sumner 
line.  Afternoon  sight  worked  as  a  Sumner  line.  Posi- 
tion of  ship  fixed  from  intersection  of  the  two  lines.  East- 
erly current  estimated.  Compass  error  again  tested. 
The  course  set  for  the  night. 

TABLES   .  153 


LIST   OF  ABBREVIATIONS 

USED  IN  THE  PRESENT  VOLUME 

Alt.  for  altitude ; 

App.  for  apparent ; 

Arg.  diff.  for  argument  difference ; 

Cf .  for  compare ; 

Chron.  for  chronometer ; 

Comp'd  for  computed ; 

Cos  for  cosine ; 

Cot  for  cotangent ; 

Csc  for  cosecant ; 

C.  —  W.  for  chronometer  minus  watch ; 
Dec.  for  declination ; 

Dep.  for  departure ; 

Dist.  for  distance ; 

D.  R.  for  dead  reckoning; 
Eq.  for  equation  of  time ; 

G.  A.  T.  for  Greenwich  apparent  time; 

G.  M.  T.  for  Greenwich  mean  time; 

Hav.  for  haversine ; 

H.  D.  for  hourly  difference; 

Int.  diff.  for  interpolation  difference ; 

Lat.  for  latitude ; 

Lat.  diff.  for  latitude  difference ; 

Log  for  logarithm ; 

Long.  for  longitude ; 

Long.  diff.  for  longitude  difference ; 

Mer.  lat.  diff.  for  meridional  latitude  difference ; 

Obs'd  for  observed ; 

p  for  polar  distance ; 

R.  A.  for  right  ascension ; 

s  for  half  sum ; 

Sec         for  secant ; 

Sin         for  sine ; 

T  for  ship's  apparent  solar  time  (or  star's  hour-angle) ; 

Tab.  diff.          for  tabular  difference ; 

Tan  for  tangent. 

xi 


NAVIGATION 

CHAPTER  I 
THE  FUNDAMENTAL  PROBLEM  OF  NAVIGATION 

To  find  one's  way  in  a  ship  across  the  trackless  ocean  is 
our  problem.  Most  people  would  like  to  know  how  it  is 
solved ;  nor  is  the  solution  very  difficult  to  understand  when 
set  forth  in  simple  language  and  without  too  great  wealth  of 
technical  detail.  We  hope  the  reader  will  find  this  to  be 
the  case  after  a  study  of  the  following  pages. 

Our  fundamental  problem  can  be  more  fully  stated  quite 
easily.  It  consists  in  the  determination  of  a  ship's  location 
on  the  earth's  surface  at  any  given  moment.  If  this  loca- 
tion can  be  determined,  it  becomes  a  comparatively  easy 
matter  to  ascertain  the  direction  (north,  south,  northeast, 
southeast,  etc.)  in  which  the  ship  must  be  steered  in  order 
to  reach  her  port  of  destination.  For  the  location  of  the 
port  of  destination  on  the  earth's 'surface  is  of  course  also 
known  :  and  if  we  know  where  the  ship  and  her  destined  port 
both  are,  we  can  easily  find  the  right  course  for  the  helmsman. 

With  the  fundamental'  problem  stated  in  this  way,  it 
would  almost  seem  as  if  there  were  really  no  such  problem 
in  existence.  For  when  the  ship  begins  her  voyage,  she  is 
necessarily  in  a  known  port.  Knowing  also  the  port  to 
which  she  is  to  go,  we  should  be  able  to  determine  her  proper 
course  from  the  one  known  port  to  the  other.  This  course 
being  then  steered,  no  further  navigational  proceedings  would 
be  required.  But  this  reasoning  is  incorrect,  because  a  ship 

B  1 


2  NAVIGATION 

does  not  actually  advance  across  the  ocean  in  exactly  the 
direction  in  which  she  is  steered.  Ocean  currents  deflect 
her ;  and  the  action  of  a  strong  wind  blowing  against  one  of 
her  sides  will  have  a  similar  effect.  Currents  and  winds 
cannot  be  predicted  with  accuracy :  and  so  it  becomes 
necessary  to  re-determine  the  ship's  position  frequently  at 
sea.  This  should  be  done  at  least  once  daily  if  possible; 
and  when  it  has  been  done,  the  mariner  can  take  a  new 
"departure,"  as  he  calls  it,  and  lay  a  new  course  for  his 
intended  port.  Thus  the  effect  of  ocean  currents,  etc.,  can 
be  eliminated,  and  the  voyage  made  as  safely  as  if  they  did 
not  exist. 

Now  this  determination  of  the  ship's  position  at  sea, 
and  when  out  of  sight  of  land,  is  strictly  an  astronomical 
problem.  It  can  be  solved  by  means  of  astronomical  ob- 
servations, and  in  no  other  way.  But  before  giving  an  out- 
line of  how  this  is  done,  let  us  first  see  what  is  meant  by 
the  words  " ship's  position  at  sea."  How  can  we  describe 
a  ship's  position  so  that  one  mariner  could  tell  another 
where  she  is  located,  and  thus  enable  the  second  mariner  to 
find  her? 

To  thus  indicate  the  point  on  the  earth's  surface  occupied 
by  the  ship  has  a  certain  similarity  with  giving  the  address  of 
a  house  in  a  city.  Such  a  city  address  always  consists  of 
two  separate  statements ;  as,  for  instance,  the  name  of  a 
street  and  the  number  of  the  house.  An  address  cannot 
be  given  completely  unless  two  different  facts  are  stated. 
They  need  not  necessarily  be  a  street  name  and  a  street 
number :  we  can  equally  well  designate  such  an  address  by 
stating  that  the  house  is  at  the  corner  of  a  certain  street  and 
a  certain  avenue.  But  here  also  the  address  is  made  up  of 
two  separate  facts. 

This  form  of  stating  an  address  as  the  intersection  of  a 
certain  street  and  avenue  is  the  form  having  the  closest 
resemblance  to  the  method  of  the  navigator.  If  the  city 
avenues  are  supposed  to  run  north  and  south,  and  the  streets 


THE   FUNDAMENTAL  PROBLEM  OF  NAVIGATION     3 

east  and  west,  as  they  do  in  New  York  (approximately),  the 
analogy  with  navigation  will  be  almost  perfect. 

For  the  navigator  imagines  the  earth  covered  with  a  net- 
work consisting  of  "  avenues, "  running  north  and  south,  and 
"  streets/'  running  east  and  west.  He  calls  the  "  avenues" 
meridians  of  longitude,  and  the  "streets"  parallels  of  latitude. 
Then  he  designates  the  position  of  a  ship  on  the  ocean  by 
stating  that  it  is  at  the  intersection  of  a  certain  meridian 
of  longitude  and  parallel  of  latitude.  There  are  360  such 
meridians  of  longitude  :  each  begins  at  the  terrestrial  equator, 
and  runs  north  and  south  from  there  to  the  north  and  south 
poles  of  the  earth.  Of  the  latitude  parallels  there  are  ISO.1 
They  all  run  east  and  west,  parallel  to  the  terrestrial  equator ; 
90  are  between  the  equator  and  the  north  pole,  and  the  other 
90  between  the  equator  and  the  south  pole. 

One  of  the  longitude  meridians  (that  passing  through 
Greenwich,  England)  is  chosen  arbitrarily  as  the  starting 
point  for  counting  longitude  meridians.  To  this  initial 
meridian  is  assigned  the  number  0,  and  the  other  meridians 
are  numbered  successively  1,  2,  3,  etc.  So  numbered, 
the  meridians  are  called  "degrees"  of  longitude;  the  third 
one,  for  instance,  being  written  3°.  The  meridians  may  be 
counted  either  eastward  or  westward  from  Greenwich,  a 
ship  on  the  20th  meridian  west  of  Greenwich,  for  instance, 
being  in  longitude  20°  west. 

The  latitude  parallels  are  similarly  counted  north  and 
south  from  the  equator ;  and  if  the  above  ship  were  on  the 
40th  latitude  parallel  north  of  the  equator,  her  complete 
"address,"  or  position  at  sea,  would  be  long.  20°  "W. ;  lat. 
40°  N. 

Of  course  a  ship  would  only  rarely  be  located  exactly  at 
the  intersection  of  a  meridian  and  parallel.  Therefore,  the 
space  between  any  two  successive  meridians  and  between 
any  two  successive  parallels  is  subdivided  into  60  parts, 
called  minutes  of  arc.  Thus  the  above  ship,  if  halfway 
1  Including  the  equator  twice,  but  excluding  the  two  poles. 


4  NAVIGATION 

between  a  pair  of  meridians  and  also  halfway  between  a 
pair  of  parallels,  might  be  in  longitude  20°  30'  west,  and 
in  latitude  40°  30'  north.  This  would  be  written  long. 
20°30'W.;  lat.  40°30'N. 

Each  minute  of  longitude  and  latitude  is  further  sub- 
divided, when  extreme  accuracy  is  required,  into  60  seconds  ; 
so  that  if  the  ship  were  a  little  to  the  north  and  a  little  to 
the  west  of  the  above  position,  she  might,  for  instance,  be 
in  long.  20°  30'  26"  W. ;  lat.  40°  30'  10"  N. 

These  meridians  and  parallels,  or  longitude  and  latitude 
lines,  appear  on  many  maps  and  charts  as  straight  lines, 
or  at  least  as  lines  only  slightly  curved.  But  being  all  lines 
imagined  drawn  on  the  earth,  which  is  almost  an  exact 
sphere  or  round  ball,  they  must  really  all  be  circles.  Thus, 
the  terrestrial  equator  is  really  a  big  circle,  girdling  the 
earth,  and  divided  into  360  equal  parts,  or.  degrees.  At 
each  of  the  division  points  a  meridian  starts  northward 
toward  the  pole.  This  meridian  is  also  a  big  circle 
perpendicular  to  the  equator.  The  distance  along  the 
meridian  from  the  equator  to  the  pole  is  divided  into  90 
equal  parts  or 'degrees,  and  the  whole  distance  from  equator 
to  pole  is  one  quarter  of  a  complete  circumference  of  the 
earth.  The  90  degrees,  from  equator  to  pole,  thus  repre- 
senting one  quarter  of  a  circumference  of  the  earth,  a  com- 
plete circumference  contains  4  X  90,  or  360  degrees,  the 
same  as  the  equator.  So  the  degrees  measured  along  the 
meridians  are  equal  to  the  degrees  measured  along  the 
equator.  The  former  are  degrees  of  latitude,  the  latter 
degrees  of  longitude;  and  degrees  of  latitude  are  equal  to 
degrees  of  longitude,  when  the  latter  are  measured  along 
the  equator.  The  length  of  each  degree  is  then  60  nautical 
miles. 

Having  thus  indicated  what  is  meant  by  a  ship's  position 
in  latitude  and  longitude,  we  shall  next  describe  in  outline 
how  such  a  position  may  be  determined  by  observation. 
If  the  ship  is  within  sight  of  a  coast-line,  there  will  probably 


THE  FUNDAMENTAL  PROBLEM  OF  NAVIGATION      5 

be  some  lighthouse,  or  other  "aid  to  navigation,"  in  view, 
from  which  the  navigator  can  ascertain  where  he  is.  Methods 
for  doing  this  are  described  later  (p.  53).  But  when  the 
ship  is  really  at  sea,  with  no  land  in  sight,  real  deep-sea 
methods  must  be  employed. 

These  methods,  when  the  weather  is  clear,  always  include 
an  observation  of  the  sun  or  some  other  heavenly  body. 
When  the  weather  does  not  permit  such  observations,  the 
mariner  can  still  find  his  position  approximately  by  means 
of  "dead  reckoning"  (abbreviated,  D.  R.).  This  process 
will  be  described  in  detail  in  the  next  chapter;  but  we  can 
already  state  that  it  consists  in  a  calculation  based  on  his 
astronomic  observation  of  latest  date.  Knowing  where  the 
ship  was  the  last  time  he  observed  the  sun,  and  also  know- 
ing both  the  direction  in  which  he  has  steered  and  the 
(approximate)  speed  of  the  ship,  the  navigator  can  calculate 
(also  approximately)  the  location  of  the  point  he  has  reached. 

Even  when  astronomical  observations  are  made,  the 
D.  R.  calculation  is  always  carried  out,  because  the  navi- 
gator is  always  anxious  to  know  how  nearly  correct  his 
D.  R.  result  would  have  been,  if  the  day  had  been  cloudy. 
Furthermore,  this  result  also  acts  as  a  check  on  the  astronomi- 
cal work,  and  tends  to  increase  the  navigator's  confidence 
in  the  correctness  of  his  final  result  as  to  the  ship's  location. 

The  manner  in  which  the  ship's  position  is  found  from 
astronomic  observations  will  of  course  be  explained  in  detail 
later.  It  is  all  done  with  an  instrument  called  a  sextant. 
This  is  merely  a  contrivance  with  which  the  navigator  can 
measure  how  high  the  sun  (or  other  heavenly  body)  is  in  the 
sky  at  any  moment.  The  sun  is  highest  in  the  sky  daily 
at  noon,  but  it  is  not  equally  high  on  different  days  in  the 
year.  Nor  is  it  equally  high  on  the  same  date  in  different 
latitudes.  Thus,  by  measuring  with  the  sextant  how  high 
it  is  on  any  particular  date  at  noon,  as  seen  from  the  ship, 
the  navigator  learns  the  terrestrial  latitude  in  which  the 
ship  is  located. 


6  NAVIGATION 

Similar  sextaht  observations  made  at  other  suitable  times 
during  the  day,  when  combined  with  exact  readings  taken 
from  an  accurate  chronometer  such  as  every  ocean-going 
ship  carries,  will  similarly  make  the  ship's  longitude  known. 
All  this  will  of  course  be  explained  in  full  detail  in  later 
chapters. 


CHAPTER  II 
DEAD   RECKONING   WITHOUT  LOGARITHMS 

As  we  have  seen  (p.  5),  this  is  a  process   by  means  of 
which  the  mariner  can  calculate  a  ship's  position  in  latitude 

West  Longitude 
61°  6.0'  59°  58°  57°  5,6°  5,5° 

46° 


Nf 
1 


45° 


44 


43 


42 


41° 


40° 


FIG.  1.  —  Dead  Reckoning.     (Diagram  not  drawn  to  scale.) 
7 


8  NAVIGATION 

and  longitude,  without  special  astronomic  observations  of 
any  kind.  In  the  accompanying  Fig.  1,  which  represents  a 
portion  of  a  chart  of  the  North  Atlantic,  a  ship's  position 
at  noon  is  shown  at  the  point  Y.  This  point  we  will  call 
the  ship's  " initial  position,"  in  discussing  our  present  prob- 
lem. We  will  suppose  that  it  was  correctly  obtained  by  as- 
tronomic observations,  and  that  these  showed  the  ship  at  Y 
to  be  in  lat.  42°  11'  N.  and  long.  59°  28'  W.  from  Green- 
wich. Sometime  in  the  afternoon,  having  traveled  a  dis- 
tance estimated  from  the  known  speed  of  the  ship  as  63  miles, 
and  having  "made  good"  this  distance  in  the  direction  YP, 
the  ship  arrives  at  P.  This  point  P  we  will  call  the  ship's 
"final  position"  ;  and  our  problem  now  is  to  find  its  latitude 
and  longitude. 

This  problem  may  be  called  the  first  fundamental  dead- 
reckoning  problem.  The  second  and  remaining  fundamental 
problem  is  the  converse  of  the  first,  and  may  be  stated  as 
follows  :  having  given  the  latitude  and  longitude  of  the  initial 
point  Y,  as  occupied  by  the  ship,  and  also  the  latitude  and 
longitude  of  the  final  point  P,  it  is  required  to  find  the  dis- 
tance from  Y  to  P  in  miles,  and  also  the  direction  of  the  line 

yp.1 

To  understand  these  two  problems  properly  it  is  next 
necessary  to  explain  how  we  may  define  the  words  "direc- 
tion YP."  This  is  done  by  referring  the  line  YP  to  the 
direction  of  the  arrow  shown  in  the  figure.  This  arrow 
is  parallel  to  the  longitude  meridians  on  the  chart,  and 
therefore  points  due  north.  The  angle  between  the  arrow 
YN  and  the  line  YP  is  marked  in  the  figure,  and  is  called 
the  "ship's  course."  This  angle  is  really  the  difference  in 
direction  of  the  two  lines  YN  and  YP.  The  point  Y  is  called 
the  "vertex"  of  the  angle,  and  all  angles  are  designated 

1  We  think  it  advisable  to  place  these  two  important  converse 
problems  together,  and  to  call  them  both  problems  of  dead  reckon- 
ing, though  many  writers  on  navigation  confine  the  phrase  "  dead 
reckoning"  to  the  first  fundamental  problem  alone. 


DEAD   RECKONING  WITHOUT  LOGARITHMS 


Dead 


by  three  letters,  the  letter  belonging  to  the  vertex  being 
placed  between  the  other  two;  in  this  case  the  angle  is 
called  either  NYP  or  PYN. 

Now  let  us  draw  a  line  PQ  (fig.  2),  from  P  to  NY,  and 
perpendicular  to  NY.  Then  the  motion  of  the  ship  from 
Y  to  P  will  have  carried  her  north  of  the 
point  Y  by  a  distance  equal  to  YQ,  and  east 
of  the  point  Y  by  a  distance  equal  to  QP.  Q 
This  is  not  strictly  true,  unless  the  earth's 
surface,  throughout  the  small  area  involved 
in  the  present  problem,  can  be  regarded 
as  a  flat  surface.  Such  a  flat  surface  is 
called  in  geometry  a  " plane"  surface;  and 
these  calculations  therefore  belong  to  that  Reckoning. 
part  of  navigation  which  is  called  " plane  sailing."  Plane- 
sailing  calculations  are  easy  calculations,  and  they  are 
generally  sufficiently  accurate  for  the  purposes  of  the 
navigator. 

The  ship's  course,  being  thus  an  angle,  must  be  designated 

by  means  of  a  unit  of  measure 
suitable  for  measuring  angles. 
For  this  purpose  the  degrees  and 
minutes  already  used  for  longi- 
tude and  latitude  (p.  3)  are 
usually  employed.  Fig.  3  shows 
that  a  latitude,  for  instance,  is 
really  an  angle,  and  must  there- 
fore also  be  measured  in  de- 
grees. P  is  the  earth's  pole,  PQ 
a  meridian,  and  the  latitude  of 

FIG.  3. -Latitude  Angle.  the  observer  at  0  IS  the  angle 

OCQ,  here  about  40°. 

So  it  is  clear  that  the  ship's  course  NYP  (figs.  1  and  2) 
will  be  measured  in  degrees.  Minutes  are  not  really  needed 
in  measuring  courses,  as  they  are  in  measuring  latitudes; 
the  nearest  whole  degree  is  always  accurate  enough,,  because 


10  NAVIGATION 

it  is  never  possible  to  steer  a  ship  on  her  proper  course  with 
absolute  exactness.  In  fact,  many  mariners  use  a  still  less 
precise  method  of  measuring  courses  by  means  of  "the  points 
of  the  compass."  (See  p.  40.) 

Resuming  our  two  fundamental  problems  (p.  8),  let  us 
now  begin  with  the  first  one,  and  proceed  to  find  the  lati- 
tude and  longitude  of  the  point  P  (figs.  1  and  2).  To  solve 
this  problem,  we  must  not  only  know  the  distance  YP 
(63  miles),  as  traveled  by  the  ship,  but  also  the  number  of 
degrees  in  the  course  angle  NYP.  Let  us  suppose  this  course 
angle  happens  also  to  be  40°.  The  problem 
then  appears  as  shown  in  Fig.  4.  We  now 
know  the  distance  YP  and  the  angle  QYP. 
Evidently  the  next  step  is  to  find  the  distances 
QFand  QP.  QY,  in  our  present  problem,  is 
called  a  "latitude  difference"  and  QP  is  called 
a  "departure." 

4. -Dead  To  find  the  "latitude  difference"  and 
"departure"  from  the  course  angle  and  dis- 
tance we  may  either  use  that  branch  of  mathematics  called 
plane  trigonometry,  or  we  may  find  them  from  a  special 
navigation  table,  called  a  "traverse  table."  Our  Table  1 
(beginning  p.  154)  is  such  a  table. 

Before x  beginning  its  use  it  will  be  well  for  the  reader  to 
note  in  general  that  all  mathematical  tables  consist  of  two 
sets  of  numbers.  The  first  set  of  numbers  are  called  "  argu- 
ments" of  the  table,  and  the  second  set  are  called  "tabular 
numbers."  The  main  object  of  the  table  is  to  furnish  us 
with  the  proper  tabular  number  when  we  know  the  proper 
argument. 

The  ordinary  multiplication  table  is  a  good  example  of  a 
mathematical  table.  It  is  usually  written  as  follows  and 

1  The  beginner  may  find  it  advisable,  on  a  first  reading  of  the 
book,  to  omit  this  explanation  of  mathematical  tables,  returning 
later  when  he  finds  a  reference  to  it  in  the  text.  The  dead  reckoning 
problem  under  discussion  is  resumed  on  p.  13. 


DEAD  RECKONING  WITHOUT  LOGARITHMS       11 


it  affords  a  good  opportunity  of  .studying  the  principles 
underlying  all  mathematical  tables  in  a  case  so  simple  as 
to  offer  no  difficulty. 

MULTIPLICATION   TABLE 

(to  illustrate  " argument "  and  "  tabular  number") 


2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

6 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48* 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

7Q 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

In  this  table  the  arguments  are  printed  in  heavy  type  and 
are  contained  in  the  left-hand  column  and  the  topmost 
horizontal  line.  In  using  the  table,  these  arguments  are 
given  in  pairs,  being  always  the  pair  of  numbers  to  be  mul- 
tiplied. In  fact,  in  the  case  of  most  tables,  the  arguments 
are  thus  given  in  pairs,  though  there  are  some  tables  with 
but  a  single  argument.  In  the  present  case  one  number 
from  the  pair  of  arguments  will  be  found  in  the  left-hand 
column,  the  other  in  the  top  horizontal  line.  Thus,  if  we  wish 
to  multiply  6  and  8,  these  two  numbers  constitute  the  pair 
of  arguments.  We  find  the  right  line  (belonging  to  6)  and 
column  (belonging  to  8) ,  and  the  tabular  number  48  (marked 
with  a  *)  occurs  at  the  intersection  of  the  6-line  and  the  8- 
column.  If  the  pair  of  arguments  are  taken  in  the  order 
8x6  instead  of  6  X  8,  we  should  use  the  8-line  and  the 
6-column,  again  finding  the  required  product  (48)  as  the 
tabular  number  at  the  intersection. 


12  NAVIGATION 

Sometimes  the  given  arguments  cannot  be  found  di- 
rectly in  the  table.  Thus  we  might  wish  to  multiply 
6|  (written  6.5)  by  8.  Evidently  the  proper  tabular 
number  would  be  halfway  between  the  6x8  tabular 
number  (48)  and  the  7  X  8  tabular  number  (56).  The 
correct  answer  would  therefore  be  52.  This  process,  by 
wh&Ii  the  tabular  number  52  is  obtained,  is  called  "  in- 
terpolation." The  example  6J  X  8  is  an  extremely  simple 
one.  When  less  easy  ones  occur,  the  interpolation  is  best 
made  as  follows :  we  ascertain  by  subtraction  how  much 
the  tabular  number  increases  while  the  argument  changes 
from  6  to  7.  This  increase  is  here  8,  because  the  tabular 
number  changes  from  48  to  56  in  the  8-column,  while  the 
argument  in  the  left-hand  column  changes  from  6  to  7. 
This  increase  of  8  in  the  tabular  number  is  called  a  "  tabular 
difference."  We  now  compare  the  given  argument  (6.5) 
with  the  nearest  argument  (6)  occurring  in  the  left-hand 
column  of  arguments,  and  find  an  " argument  difference" 
of  0.5  (being  6.5  minus  6).  Since  this  "  argument  dif- 
ference" is  0.5,  we  must  evidently  take  0.5  X  8  (8  being  the 
tabular  difference),  and  increase  the  tabular  number  48  by 
0.5  X  8,  or  4.  This  again  brings  us  to  52.  Similar  exam- 
ples are : 

(1)  5.3  X  4  =  21.2;    (2)  7.7  X  8  =  61.6. 

In  example  (1)  the  tabular  numbers  are  20  and  24;  the 
tabular  difference  is  4.  0.3  X  4  =  1.2;  20  +  1.2  =  21.2,  the 
answer.  Both  examples  may  be  verified,  of  course,  by  ordi- 
nary multiplication. 

When  both  given  arguments  contain  fractions,  as,  for 
instance,  5.3  X  8.4,  the  resulting  " double  interpolation" 
is  so  complicated  as  to  be  of  little  practical  use  to  the  navi- 
gator. 

To  make  this  general  explanation  of  mathematical  tables 
complete,  it  remains  to  show  how  they  can  be  used  in  an 
inverse  manner ;  i.e.  to  find  the  argument  from  the  tabular 


DEAD  RECKONING  WITHOUT   LOGARITHMS       13 

number.  Thus,  if  we  were  told  that  the  tabular  number  is 
48,  and  one  argument  8,  an  inspection  of  the  table  would 
at  once  show  that  the  other  argument  must  be  6.  In  this 
way  the  table  might  be  used  for  division  as  well  as  multi- 
plication ;  and  interpolation  would  evidently  also  be  possible. 
Many  mathematical  tables  must  frequently  be  thus  used 
in  an  inverse  manner. 

Having  thus  explained  the  peculiarities  of  mathematical 
tables,  we  return  to  our  dead-reckoning  problem  and  its 
solution  by  means  of  the  traverse  table  (p.  154). 

Referring  to  that  table  we  find  a  column  (p.  167), 
headed  40°,  the  course  angle  of  our  present  problem.  On 
the  left-hand  side  of  the  page  we  find  the  given  distance,  63. 
Then,  opposite  the  distance  63,  and  under  40°,  we  find  the 
latitude  difference  (abbreviated,  "Lat.")  and  the  departure 
(abbreviated,  "Dep.")  to  be: 

lat.  =  48.3,  dep.  =  40.5. 

The  following  are  additional  examples  for  practice : 

Given :  dist.,  84,  course  26° ;  Arcs.,  lat.  =  75.5,  dep.  =  36.8. 
Given :  dist.,  28,  course  11° ;  Arcs.,  lat.  =  27.5,  dep.  =  5.3. 

When  the  course  is  between  1°  and  45°  the  course  angle 
will  be  found  in  Table  1  at  the  head  of  the  column  :  but  when 
the  course  is  between  45°  and  90°,  it  appears  at  the  foot  of 
the  column.  In  the  latter  case,  the  tabular  lat.  and  dep. 
are  to  be  taken  from  the  columns  having  "Lat."  and  "Dep." 
at  the  foot  instead  of  the  top  of  the  column.  Examples 
follow :  « 

Given :  dist.,  63,  course  50° ;  Arcs.,  lat.  =  40.5,  dep.  =  48.3. 
Given :  dist.,  84,  course  64° ;  Arcs.,  lat.  =  36.8,  dep.  =  75.5. 
Given :  dist.,  28,  course  52°  ;  Arcs.,  lat.  =  17.2,  dep.  =22.1, 

In  addition  to  the  course  angles  from  1°  to  90°,  three  ad- 
ditional angles  are  given  in  parentheses  at  the  top  and  foot 
of  each  column.  Thus,  with  the  course  angle  30°  appear 
also  150°,  210°,  330°.  This  simply  means  that  the  latitudes 


14 


NAVIGATION 


and  departures  are  the  same  for  these  four 
course  angles.  The  accompanying  Fig.  5  shows, 
for  instance,  that  the  departures  QP  and  Q'P' 
are  equal  for  30°  and]  150°  courses  if  the  two 
distances  YP  and  YP'  are  alike. 

It  will  be  noticed  also  that  our  traverse  table 
always  gives  distances  from  1  to  50  on  a  left- 
hand  page,  and  from  50  to  100  on  a  right-hand 
page.  When  distances  larger  than  100  occur, 
it  is  necessary  to  use  the  100,  200,  etc.,  given  on 
the  lower  part  of  each  page.  If,  for  instance, 
we  require  the  latitude  and  departure  for  a 
distance  363  miles,  course  40°,  we  turn  again  to 
.the  40°  column,  and  find  (near  the  bottom  of 
30°  and  150°.  the  page) : 

For  300  miles,  lat.  =  229.8,  dep.  =  192.8 
and  (in  the  usual  way)  for  63  miles,  lat.  =   48.3,  dep.  =   40.5 
Sums,  363  =278.1  233.3 

Consequently,  for  dist.  363,  course  40°,  lat. =278.1,  dep. =233.3. 

Other  examples  are : 

Course  25°,  dist.,  452 ;  lat.  =  409.6,  dep.  =  191.0. 
Course  68,  dist.,  521 ;  lat.  =  195.2,  dep.  =  483.1. 
Course  226,  dist.,  384 ;  lat.  =  266.8,  dep.  =  276.2. 

When  the  given  distances  or  course  angles,  which  are 
really  the  "pairs  of  arguments"  (p.  11)  of  the  traverse  table, 
contain  fractions,  interpolation  can  be  used ;  but  such  close 
accuracy  is  seldom,  if  ever,  required  in  navigation. 

More  extended  traverse  tables  will  be  found  in  Bowditch's 
"American  Practical  Navigator,"  published  by  the  Navy 
Department,  Washington.  They  are  also  printed  separately 
in  Bowditch's  "Useful  Tables."  Both  volumes  can  be 
purchased  at  any  "navigation  shop "  where  instruments 
and  books  suitable  for  navigators  are  sold. 

To  complete  this  explanation  of  our  traverse  table,  it  is 
still  necessary  to  mention  that  it  also  provides,  with  suf- 
ficiently close  approximation,  for  the  method  of  measuring 


DEAD  RECKONING  WITHOUT  LOGARITHMS       15 

course  angles  in  " points  of  the  compass"  (pp.  10,  41).  This 
method  is  not  now  in  use  in  the  United  States  Navy,  but  it 
is  still  largely  employed  in  merchant  vessels.  It  is  sufficient 
to  state  here  that  a  course  of  3  points,  for  instance,  is  very 
nearly  equal  to  a  course  of  34°,  and  the  traverse  table  column 
for  34°  may  properly  be  used  for  a  3-point  course.  Similarly, 
31°  may  be  used  for  2|  points,  and  the  mariner  desiring  to  use 
points  can  always  find  from  the  traverse  table  itself  just 
what  column  to  use.  A  special  traverse  table  for  points  may 
also  be  found  in  Bowditch's  Tables,  already  mentioned. 

We  have  now  shown  how  to  find  latitude  difference  and 
departure  by  means  of  the  traverse  table.  But  our  problem 
is  not  yet  completely  solved.  Our  ship  (p.  8)  started  from 
the  point  7  in  lat.  42°  11'  N. ;  long.  59°  28'  W.  She  traveled 
63  miles  on  a  40°  course,  and  the  traverse  table  showed  that 
she  thus  made  good  a  latitude  difference  of  48.3  miles  and  a 
departure  of  40.5  miles.  It  now  remains  to  ascertain  how 
much  the  ship  changed  her  latitude  in  degrees  and  minutes 
from  42°  11'  N.  and  her  longitude  in  degrees  and  minutes 
from  59°  28'  W.  When  we  have  found  these  last  changes, 
we  can  learn  the  latitude  and  longitude  of  the  point  P, 
which  we  are  required  to  find. 

To  get  the  latitude  change  in  degrees  and  minutes  from 
the  latitude  difference  in  miles  offers  no  difficulty.  If  the 
miles  used  are  nautical  miles  (and  in  navigation  they  always 
are  nautical  miles),  each  mile  of  latitude  difference  corre- 
sponds to  1'  of  angular  measure  (p.  9),  and  60  miles  corre- 
spond to  1°.  Thus  our  ship  must  have  changed  her  latitude 
48'.3,  corresponding  to  a  latitude  difference  of  48.3  miles. 
Her  initial  latitude  having  been  42°  11'  N.,  her  final  latitude 
at  P  will  be  42°  11'  +  48'  (if  we  omit  the  odd  .3)  or  42°  59'  N. 

The  relation  between  departure  and  difference  of  longitude 
is  not  quite  so  simple.  Our  ship's  departure  of  40.5  miles 
might  correspond  to  far  more  than  40.5  minutes  of  longitude. 
In  fact,  in  very  high  latitudes  near  the  north  pole,  the  longi- 
tude meridians  converge  so  closely  that  a  person  traveling 


16  NAVIGATION 

a  few  miles  might  change  his  longitude  very  greatly.  At  the 
pole  itself  a  man  might  change  his  longitude  180°  by  simply 
stepping  across  the  pole.  So  it  follows  that  the  longitude 
difference  in  minutes  is  greater  than  the  departure  in  miles 
(however,  cf.  p.  4).  The  difference  between  the  two  increases 
rapidly  as  we  approach  high  latitudes  though  it  is  nil  at 
the  equator;  in  Table  2  (beginning  p.  168)  we  give  this 
excess  of  longitude  difference  over  departure  for  all  latitudes 
under  60°,  and  for  all  longitude  differences  up  to  100.  When 
the  longitude  differences  are  greater  than  100J  it  is  necessary 
to  use  the  numbers  given  for  100,  200,  300,  etc.,  near  the 
bottom  of  each  page  in  the  table,  and  to  sum  tabular  num- 
bers, precisely  as  we  did  with  the  traverse  table. 

It  will  be  noticed  that  Table  2  gives  "tabular  numbers" 
for  each  degree  of  latitude  in  a  separate  column,  and  that 
these  various  latitudes  are  called  "middle  latitudes."  Thus 
the  middle  latitude  and  the  longitude  difference  are  the  pair 
of  arguments  (p.  11)  for  Table  2,  and,  as  we  shall  see  pres- 
ently, the  use  of  the  middle  latitude  avoids  any  uncertainty 
in  choosing  the  correct  column  for  use.  In  our  present 
problem  we  have  at  our  disposal  (p.  15)  two  different  lat- 
itudes :  the  initial  latitude  at  the  point  Y,  42°  11'  N.,  and 
the  final  latitude  at  the  point  P,  42°  59'  N.  In  this  case,  the 
two  latitudes  are  so  nearly  equal  that  we  might  use  either 
of  them  as  an  argument  in  Table  2  without  material  inaccu- 
racy. In  fact,  in  using  Table  2  it  is  unnecessary  to  consider 
minutes  of  latitude,  the  nearest  degree  being  sufficient. 

But  often  the  two  latitudes  available  at  this  stage  of  the 
problem  differ  by  many  degrees.  In  such  cases  mariners 
always  use  the  average  of  the  two  latitudes,  and  call  it  the 
"middle  latitude."  In  the  present  case,  the  middle  latitude 
would  be  found  thus : 

Initial  latitude  =  42°  11' 

Final  latitude  =  42°  59' 

Sum  =  85°  10' 

sum  =  middle  latitude  =  42°  35' 


DEAD  RECKONING  WITHOUT  LOGARITHMS       17 

The  nearest  even  degree  to  42°  35'  is  43°,  and  the  prob- 
lem would  therefore  be  worked  with  the  43°  column  of  middle, 
latitude  in  Table  2. 

Before  completing  our  problem  it  is  necessary  to  point 
out  that  while  Table  2  is  intended  primarily  for  changing 
longitude  differences  in  minutes  into  departures  in  miles,  it 
can  also  be  used  (as  stated  at  the  foot  of  each  page)  for  the 
inverse  transformation  of  departures  into  longitude  dif- 
ferences ;  and  this,  is  the  transformation  we  must  make  in 
our  present  problem.  It  is  merely  necessary  to  use  the 
departure  (40.5)  in  the  left-hand  column,  at  the  head  of 
which  are  the  words  "Long.  Diff.  or  Dep.,"  indicating  that 
either  of  these  two  may  be  used  as  the  argument  in  that 
column.  Then,  in  the  43°  column  of  middle  latitude,  we 
find  (using  interpolation)  the  tabular  number  10.8. 

This  means  that  a  longitude  difference  of  40'. 5  corre- 
sponds to  a  departure  of  40.5  —  10.8  miles,  or  29.7  miles. 

But  when  the  table,  as  in  the  present  case,  is  used  for  the 
inverse  transformation,  the  tabular  number  10.8  must, 
before  use,  be  multiplied  by  the  factor  given  at  the  bottom 
of  the  column.  For  the  middle  latitude  43°  this  factor  is 
1 .37 ;  and  so  the  right  tabular  number  becomes,  in  the 
present  case : 

10.8  X  1.37  =  14.8; 

and  as  the  longitude  difference  is  always  greater  than  the 
departure,  it  follows  that  the  departure  of  40.5  miles  gives 
a  longitude  difference  of : 

40.5  +  14.8  =  55'.3  =  0°  55', 

if  we  omit  the  odd  tenths. 

The  initial    longitude  of  the  ship  at  the  point  Y  was 
59°  28'  W.     As  her  40°  course  has  carried  her  nearer  to  Green- 
wich, it  follows  that  her  final  longitude  at  the  point  P  is : 
59°  28'  W.  -  0°  55'  =  58°  33'  W. 

We  shall  now  discuss  the  following  similar  problem : 

A  ship  takes  her  departure  from  a  point  about  one  mile 


18  NAVIGATION 

east  of  Navesink  Highlands  Light,  New  Jersey,  in  the  initial 
lat.  40°  24'  N.,  initial  long.  73°  58'  W.,  and  travels  1377 
miles  on  a  course  of  166°.  What  final  latitude  and  longitude 
does  she  attain? 

Entering  the  traverse  table  in  the  column  headed  166°, 
which  is  the  same  as  the  14°  column,  we  find  : 

For  dist.  900,  lat.,    873.2,  dep.,  217.7 

For  dist.  400,  lat.,    388.1,  dep.,    96.7 

For  dist.  77,  lat.,      74.7,  dep.,    18.6 

Sums,  1377,          1336.0,            333.0 

To  make  the  large  given  distance  (1377  miles)  come  within 
the  range  of  Table  1,  it  has  been  necessary  to  enter  the  166° 
column  three  times,  with  the  arguments  900,  400,  and  77, 
and  then  to  sum  the  corresponding  tabular  numbers. 

The  latitude  difference,  1336  miles,  is  equivalent  to  1336', 
or  22°  16',  counting,  as  usual,  60'  to  1°.  Then,  since  the 
direction  of  her  course  (166°)  carried  the  ship  to  the  south 
of  her  initial  position  (cf .  Fig.  5,  p.  14,  and  p.  19),  we  have : 

Initial  lat.,  40°  24'  N. 
Lat.  diff.,  22°  16'  N. 
Final  lat.,  18°  8' N. 
Middle  lat.,  29°  16'  N. 

Now  turning  to  Table  2,  in  the  proper  column  for  middle 
latitude  29° : 

For  dep.  300  tabular  number  is  37.6 
For  dep.  33  tabular  number  is  4.1 
Sums  333  41.7 

As  in  the  former  example,  this  41.7  must  be  multiplied 
by  the  factor  at  the  bottom  of  the  column.  This  factor  is 
1.14.  Multiplying,  we  have:  41.7  X  1.14  =  47.5.  Conse- 
quently, long.  diff.  =  333  +  47.5  =  380'.5  =  6°  20'.5.  Since 
the  direction  of  her  course  (166°)  carried  the  ship  eastward, 
and  therefore  nearer  to  Greenwich,  it  follows  that  her  final 
longitude  is  73°  58'  W.  -  6°  20',  or  67°  38'  W.  The  final 
position  is  therefore  :  lat.  18°  8'  N. ;  long.  67°  38'  W. 


DEAD  RECKONING  WITHOUT  LOGARITHMS       19 

The  point  indicated  by  this  final  latitude  and  longitude 
is  just  off  tjie  entrance  to  the  Mona  Passage,  between  Haiti 
and  Porto  Rico ;  the  given  course  and  distance  would  there- 
fore be  correct  for  a  voyage  from  New  York  to  Mona  Passage. 

Additional  similar  problems  are  : 

1.  Initial  lat.,  40°  28'  N. ;  initial  long.,  73°  50'  W. ;  course, 
119°;  dist.,  2924  miles.     This  would  take  the  ship  from 
Sandy  Hook  to  St.  Vincent,  Cape  Verde  Islands. 

Ans.  Final  lat.,  16°  50'  N. ;  final  long.,  25°  7'  W. 

2.  Initial  lat.,  40°  W  N. ;  initial  long.,  70°  0'  W. ;  course, 
75° ;  dist.,  2606  miles.     This  would  take  the  ship  from  Nan- 
tucket  Lightship  to  Fastnet,  the  nearest  point  of  the  Irish 

coast. 

Ans.  Final  lat.,  51°  24'  N. ;  final  long.,  9°  37'  W. 

Before  proceeding  to  our  second  fundamental  problem 
(p.  8),  it  will  be  well  to  explain  briefly  two  further  points 
of  interest.  The  first  of  these  relates  to  the  method  of  desig- 
nating a  ship's  course.  We  have  hitherto  supposed  it  to 
be  measured  in  degrees,  from  the  north,  around  by  way  of 
the  east,  through  the  south  and  west,  and  so  back  to  the 
north  again.  This  is  the  best  way  to  count  courses,  and  is 
the  way  now  in  use  in  the  United  States  Navy.  Since  a 
whole  circle  contains  360°,  it  follows  that  courses  may  con- 
tain any  number  of  degrees  from  0°  to  360°. 

But  there  is  another  quite  convenient,  although  older,  way 
of  designating  courses,  in  which  a  60°  course,  for  instance,  is 
written  N.  60°  E.,  showing  that  the  ship  must  be  steered  60° 
east  of  north.  In  a  similar  way,  a  120°  course  is  written 
S.  60°  E.,  showing  that  the  helmsman  should  head  her  60° 
east  of  south,  which  would  be  the  same  as  30°  south  of  east, 
or  120°  from  the  north  toward  the  south  by  way  of  east. 

The  second  further  point  of  interest  has  to  do  with  the 
relation  between  Tables  1  and  2.  It  is  possible  to  avoid 
entirely  the  use  of  Table  2,  and  to  transform  longitude  differ- 
ences into  departures,  and  vice  versa,  by  means  of  Table  1 


20  NAVIGATION 

alone.  It  so  happens  that  the  relation  between  these  two, 
for  any  given  middle  latitude,  as,  for  instance,  §9°,  is  iden- 
tical with  the  relation  between  distance  and  latitude  difference 
in  Table  1  for  the  course  29°.  In  other  words,  if  we  have 
given  a  middle  latitude  and  a  longitude  difference,  and  wish 
to  find  the  departure,  we : 

Call  the  middle  latitude  a  course,  and 
Call  the  longitude  difference  a  distance ; 

Then,  corresponding  to  that  course  and  distance,  find  from 
Table  1  the  tabular  latitude  difference,  and  it  will  be 
the  required  departure.  The  same  process  can  also  be 
reversed,  so  as  to  find  the  longitude  difference  from  the 
departure. 

While  this  method  with  Table  1  is  quite  correct,  we  believe 
beginners  (at  least)  will  find  the  use  of  Table  2  advantageous 
in  the  solution  of  these  problems,  especially  when  the  middle 
latitude  is  not  very  great. 

Coming  now  to  our  second  fundamental  problem  of  dead 
reckoning,  let  us  suppose  a  ship  is  required  to  proceed  from 
the  initial  lat.  42°  11'  N.  and  long.  59°  28'  W.  to  a  final 
lat.  42°  59'  N.  and  long.  58°  33'  W.  We  are  to  find  the  course 
she  must  steer,  and  the  distance  she  must  run. 

We  have  at  once  the  latitude  difference  of  0°  48',  or  48  miles, 
and  the  middle  latitude  42°  35',  or  nearest  whole  degree  of  mid- 
dle latitude,  43° .  The  longitude  difference  is  55' ;  and  with  this 
•we  find  from  Table  2  the  correction  14.8  in  the  43°  column 
of  middle  latitude.  Remembering  that  this  time  we  are 
transforming  a  longitude  difference  into  departure,  and  con- 
sequently do  not  need  to  use  the  factor  at  the  foot  of  the 
column,  we  subtract  this  correction  (14.8)  from  the  longi- 
tude difference  (55')  and  obtain  the  departure  as  40.2  miles. 

Next  we  proceed  to  Table  1,  to  find  the  course  and  distance 
corresponding  to  lat.  48,  dep.  40.2.  To  do  this,  we  must 
find  a  place  in  Table  1  where  this  particular  latitude  and 
departure  appear  side  by  side.  If  this  pair  of  numbers 


DEAD  RECKONING  WITHOUT   LOGARITHMS      21 

cannot  be  found  (exactly)  side  by  side,  we  must  take  the 
pair  which  come  nearest  to  them :  in  this  case  such  a  pair 
of  numbers  is  found  in  the  40°  course  column,  opposite  dist. 
63.  So  it  appears  that  the  ship  must  steer  on  a  40°  course 
a  distance  of  63  miles,  to  proceed  from  the  given  initial  to 
the  given  final  latitude  and  longitude.  This  problem  is  the 
direct  converse  of  the  one  first  solved  (pp.  15,  17). 

As  a  second  example,  let  us  now  calculate  the  course  and 
distance  from  Sandy  Hook,  lat.  40°  28'  N. ;  long.  73°  50'  W., 
to  St.  Vincent,  lat.  16°  50'  N. ;  long.  25°  7'  W.  We  have, 
by  subtraction,  lat.  diff.  =  23°  38'  =  1418'  =  1418  miles; 
long.  diff.  =  48°  43'  =  2923'. 

This  2923'  must  be  turned  into  a  departure,  the  middle 
latitude  being  28°  39',  or,  to  the  nearest  whole  degree,  29°. 
Turning  to  the  column  of  Table  2  which  belongs  to  29°  of 
middle  latitude,  we  find  the  correction  for  2923'  of  longitude 
difference  thus : 

Tabular  number  for  900  =  113.0, 

which  being  multiplied  by  3,  gives : 

Tabular  number  for  2700  =  339.0 

Also,  tabular  number  for       200=    25.1 
Tabular  number  for  23  =      2.9 

Sums,  tabular  number  for  2923  =  367.0 

This  must  be  subtracted  from  the  longitude  difference,  and 

so  we  get : 

dep.  =  2923  -  367.0  =  2556  miles. 

We  have  now  to  seek  a  place  in  Table  1  where  lat.  1418  and 
dep.  2556  appear  side  by  side.  No  traverse  tables  are  suffi- 
ciently extended  to  contain  these  large  numbers,  but  we 
.  can  at  once  obtain  an  approximate  answer  to  the  problem 
by  dividing  both  numbers  by  100.  This  reduces  them  to 
lat.  14.2,  dep.  25.6  ;  and  the  nearest  numbers  to  these  which 
can  be  found  side  by  side  in  Table  1  are  in  the  column  belong- 
ing to  course  119°  and  opposite  dist.  29.  This  course  (119°) 
is  the  same  as  would  have  been  obtained  if  we  had  not  been 


22  NAVIGATION 

forced  to  divide  our  latitude  and  departure  by  100,  to  bring 
them  within  the  range  of  Table  1.  But  the  dist.  29  must 
now  be  multiplied  by  100,  to  remove  the  effect  of  our  former 
division  of  latitude  and  departure  by  100.  Thus  we  have 
the  closely  approximate  information  that  the  course  and 
distance  from  Sandy  Hook  to  St.  Vincent  are  119°  and  2900 
miles.  The  same  problem  (p.  19),  when  taken  in  its  inverse 
form,  starts  with  the  numbers  119°  and  2924  miles. 

In  discussing  such  a  problem,  many  beginners  have  dif- 
ficulty in  choosing  correctly  the  course  number  (119°)  from 
the  four  (61°,  119°,  '241°,  299°)  to  be  found  at  the  foot  of 
the  same  column  of  Table  1 .  This  choice  is  easily  made  with 
the  help  of  our  knowledge  of  elementary  geography,  or  with 
any  rough  chart  or  map.  From  these,  we  know  that  St. 
Vincent  is  south  and  east  of  Sandy  Hook,  and  the  only  one 
of  the  four  possible  courses  that  will  carry  a  ship  south  and 
east  is  course  119°.  The  same  course  might  be  written  in 
the  other  notation  (p.  19)  S.  61°  E.,  which  possibly  makes 
the  actual  direction  to  be  steered  a  little  easier  to  under- 
stand. 

The  above  result  is  approximate  only,  but  higher  accuracy 
is  seldom  required.  When  desired,  it  can  be  obtained  by 
certain  kinds  of  interpolations  (p.  12) ;  but  these  are  always 
unsatisfactory,  especially  as  complete  precision  can  always 
be  easily  had  by  the  use  of  logarithms,  as  explained  in  the 
next  chapter. 


CHAPTER  III 
DEAD  RECKONING  WITH   LOGARITHMS 

SINCE  the  publication  in  1876  of  Kelvin's  tables  for 
facilitating  Sumner's  method,  it  has  been  possible  to  navi- 
gate in  the  most  approved  way  without  using  logarithms  or 
trigonometry.  Those  who  desire  to  study  the  subject  in 
this  manner  may  do  so  by  simply  omitting  those  parts  of 
the  book  in  which  logarithmic  or  trigonometric  formulas 
and  calculations  occur.  But  this  method  of  study  is  not 
recommended,  except  perhaps  for  a  first  reading;  for  a 
knowledge  of  logarithmic  processes  always  affords  a  most 
desirable  check  on  the  accuracy  of  the  other  method,  and 
so  makes  for  safety  of  the  ship  and  peace  of  mind  of  the 
navigator. 

Proceeding,  then,  with  the  subject  of  logarithms,  we  may 
define  them  as  a  mathematical  device  for  facilitating  calcula- 
tions. They  are  merely  numbers ;  but  they  are  numbers 
having  this  peculiarity :  every  logarithmic  number  belongs 
to  some  ordinary  number  (like  1,  2,  3,  27,  800,  etc.),  and 
belongs  to  it  alone.  Its  logarithm  belongs  to  the  number  as 
a  man's  shadow  belongs  to  the  man. 

For  our  present  purpose  it  is  unnecessary  to  enter  into  the 
theory  of  logarithms ;  we  shall  explain  only  the  methods  of 
using  them  in  practice.  Logarithms  (abbreviated  "log") 
always  consist  of  two  parts,  a  "whole  number"  part  and  a 
"decimal"  part.  Thus,  3.30103  is  a  logarithm,  of  which 
the  whole  number  part  is  3,  and  the  decimal  part  .30103. 
The  whole  number  part  may  even  be  zero :  thus,  0.30103 
is  also  a  logarithm.  The  decimal  part  of  the  logarithm 
is  found  from  a  table  of  logarithms,  such  as  our  Table  3 

23 


24  NAVIGATION 

(p.  178) ;  but  the  whole  number  part  is  found  by  an  inspec- 
tion of  the  number  to  which  the  logarithm  belongs. 

We  shall  hereafter,  to  save  space,  always  write  "log  26" 
in  place  of  "the  logarithm  belonging  to  26":  and,  with 
the  help  of  this  abbreviation,  we  may  now  write  the  follow- 
ing tabular  statement,  which  is  fundamental  in  the  matter 
of  logarithms  : 

log  1  =  0.00000,  log  1000  =  3.00000, 
log  10  =  1.00000,  log  10000  =  4.00000, 
log  100  =  2.00000,  log  100000  =  5.00000,  etc. 

In  other  words,  for  these  particular  numbers,  all  "mul- 
tiples" of  10,  the  decimal  part  of  the  log  is  zero.  For 
numbers  intermediate  between  1  and  10,  the  whole  number 
part  of  the  log  is  0,  and  the  decimal  part  lies  between 
.00000  and  .99999.  For  those  between  10  and  100  the  whole 
number  part  is  1,  and  the  decimal  part  again  lies  between 
.00000  and  .99999. 

The  general  rule  is :  the  whole  number  part  of  a  log  is 
one  less  than  the  number  of  figures  or  "digits"  in  the  number 
to  which  the  log  belongs.  Thus,  the  number  26  has  two 
digits  :  the  whole  number  part  of  its  log  is  1.  The  number 
2678  has  four  digits :  the  whole  number  part  of  its  log  is 
therefore  3. 

If  a  number  is  itself  partly  decimal,  we  count  only  the 
number  of  digits  to  the  left  of  the  decimal  point  for  the  pur- 
poses of  the  present  rule.  Thus,  26.78  has  two  digits  only ; 
2.678  has  one ;  267.8  has  three,  etc. 

If,  on  the  other  hand,  a  number  is  wholly  decimal,  as 
0.2678,  the  whole  number  part  of  its  logarithm  should  be 
"negative,"  or  minus,  i.e.  less  than  0;  and  it  will  be  one 
greater  than  the  number  of  zeros  immediately  following  the 
decimal  point  in  the  number.  According  to  this,  the  whole 
number  part  of  log  0.2678  should  be  —  1,  because  this 
number  has  no  zeros  immediately  following  the  decimal 
point.  But  as  these  negative  whole  number  parts  are 
very  inconvenient  in  actual  work,  it  is  customary  to  increase 


DEAD  RECKONING  WITH  LOGARITHMS  25 

all  logs  of  decimal  numbers  arbitrarily  by  10,  which  will 
avoid  the  negative  sign.  This  arbitrary  increase  is  always 
corrected  again  in  the  further  or  final  procedure,  so  that  it 
cannot  possibly  introduce  error  into  the  work. 

In  the  case  of  log  0.2678,  the  arbitrary  increase  of  10 
changes  the  —  1  to  +  9 l ;  and  so  9  would  be  the  whole 
number  part  of  log  0.2678.  Similarly,  log  0.002678  would 
have  7  for  its  whole  number  part,  because  there  are  two  zeros 
after  the  decimal  point.  This  would  make  the  whole  number 
part  of  the  log  —  3,  which,  being  increased  by  10,  gives  +  7. 

In  general,  this  matter  of  logs  of  wholly  decimal  numbers 
may  be  summarized  as  follows : 

log  0.1  =9.00000,  log  0.0001  =6.00000, 
log  0.01  =8.00000,  log  0.00001  =5.00000, 
log  0.001  =  7.00000,  log  0.000001  =  4.00000,  etc. 

In  all  these  cases  the  decimal  part  of  the  log  is  zero: 
and  if  the  number  lies,  for  instance,  between  0.1  and  0.01, 
the  whole  number  part  of  the  log  will  be  8,  and  the  decimal 
part  will  lie  between  .00000  and  .99999. 

The  decimal  part  in  the  log  of  any  number  is  taken  from 
Table  3  without  regard  to  the  position  of  the  decimal 
point  in  the  number  itself.  The  numbers  0.2678,  0.002678, 
26.78,  2.678,  267.8,  and  2678  all  have  precisely  the  same 
decimal  part  in  their  logs,  so  that  such  logs  will  differ  in 
their  whole  number  parts  only.  We  can  at  once  obtain  this 
common  decimal  part  from  Table  3  (p.  181),  where  it  is 
found  to  be  .42781.  In  looking  up  this  log,  we  again  use 
(p.  11)  a  pair  of  arguments.  The  argument  for  the  left- 
hand  column  consists  of  the  first  three  digits  of  2678  (267) ; 
and  in  selecting  this  argument  we  disregard  any  zeros  that 
may  immediately  follow  the  decimal  point,  if  the  number 
is  wholly  decimal,  like  .002678.  The  other  argument,  in 
the  top  horizontal  line  of  the  tabular  page  is  8,  the  right- 
hand  digit  of  the  number  2678.  In  the  horizontal  line 

1  According  to  Algebra,  9  is  greater  than  -  1  by  10. 


26  NAVIGATION 

opposite  267,  and  in  the  column  headed  8,  appears  781 ;  and 
these  are  the  last  three  digits  of  the  required  log  (.42781). 
The  first  two  digits  (.42)  are  common  to  a  great  many  logs, 
and  are  therefore  only  printed  in  the  column  headed  0. 
The  first  two  digits  of  every  log  are  thus  taken  from  the 
zero  column,  regularly  from  the  same  horizontal  line  that 
contains  the  last  three  digits  of  the  log,  or  from  some  line 
above  it.  Only  when  there  is  an  asterisk  printed  in  the  table 
with  the  last  three  digits  do  we  make  an  exception,  and  take 
the  first  two  digits  from  the  line  below  the  one  containing  the 
last  three.  Thus  the  decimal  part  of  log  2691  is  .42991,  but 
the  decimal  part  of  log  2692  is  .43008. 

Having  thus  found  the  decimal  part  of  log  2678  to  be 
.42781,  and  the  number  2678  having  four  digits,  the  com- 
plete 

log  2678  =  3.42781 ; 

and  here  the  reader  should  once  more  note  that  all  tabular 
logs  like  .42781  are  thus  always  decimals.  The  correspond- 
ing logs  for  the  other  numbers  given  above  are : 

log  267.8  =  2.42781, 
log  26.78  =  1.42781, 
log  2.678  =  0.42781, 
log  0.2678  =  9.42781, 
log  0.002678  =  7.42781. 

It  is  clear  that  Table  3  gives  directly  the  decimal  part  of 
the  logs  of  all  numbers  containing  four  digits.  If  the  number 
contains  less  than  four  digits,  as  26,  we  should  look  it  up  in 
the  table  as  if  it  were  2600.  We  should  find  260  as  the 
argument  in  the  left-hand  column  (p.  181) ;  and  in  the 
corresponding  line,  in  the  column  headed  0  (the  fourth  digit 
of  2600),  is  41497.  This  is  the  decimal  part,  as  usual,  and 
the  complete 

log  26  =  1.41497. 

If,  on  the  other  hand,  the  number  whose  log  is  wanted 
contains  more  than  four  digits,  as  26782,  it  is  necessary  to 


DEAD  RECKONING  WITH  LOGARITHMS          27 

resort  to  interpolation  (p.  12).  The  number  of  digits  being 
here  5,  the  whole  number  part  of  the  log  is  4  (p.  24).  The 
decimal  part  of  the  log  is  to  be  found  quite  without  regard 
to  decimal  points  (p.  25).  It  may  therefore  be  taken 
from  Table  3  just  as  if  we  wanted  log  2678.2  instead  of  26782. 
Now  the  table  tells  us  (p.  181) : 

decimal  part  of  log  2678  =  42781, 
decimal  part  of  log  2679  =  42797. 

The  tabular  difference  (p.  12)  of  these  two  decimal  parts 
is  16.  As  26782  may,  for  our  present  purpose,  be  regarded 
as  lying  &  of  the  way  from  2678  to  2679,  it  follows  that  the 
decimal  part  of  log  26782  will  lie  T2^  of  the  way  from  42781 
to  42797.  Evidently,  we  must  multiply  the  tabular  differ- 
ence 16  by  ^  (giving  3.2)  to  find  how  much  larger  the  decimal 
part  of  log  26782  is  than  the  decimal  part  of  log  2678. 
This  3.2  (or  3,  in  round  numbers)  must  then  be  added  to 
42781 ;  and  we  have,  as  the  result  of  this  interpolation : 

decimal  part  of  log  26782  =  .42784. 

As  we  have  just  found  the  whole  number  part  to  be  4, 
we  have  for  the  complete  : 

log  26782  =  4.42784. 

This  whole  process  of  interpolation  may  perhaps  be  more 
clearly  understood  if  we  repeat  (p.  10)  that  all  tables  furnish 
tabular  numbers  corresponding  to  given  arguments.  In- 
terpolation is  necessary  when  the  given  arguments  are  not 
to  be  found  in  the  argument  part  of  the  table,  but  fall 
between  two  of  the  tabular  arguments.  Then  we  obtain 
by  subtraction  the  difference  between  the  given  argument 
and  the  nearest  smaller  argument  contained  in  the  table. 
This  difference  is  the  " argument  difference"  (abbreviated, 
arg.  diff.),  and  it  should  be  expressed  as  a  decimal  fraction 
of  the  interval  between  two  successive  arguments  (cf .  T%, 
above).  The  tabular  difference  (tab.  diff.)  between  two 
successive  tabular  numbers  being  also  obtained  by  subtrac- 


28  NAVIGATION 

tion,  we  have  only  to  multiply  the  tabular  difference  by  the 
argument  difference  to  find  the  "interpolation  difference" 
(int.  diff.).  This  is  then  added  1  to  the  proper  tabular 
number  (belonging  to  the  above-mentioned  nearest  argu- 
ment given  in  the  table)  to  obtain  the  tabular  number  re- 
quired. 

The  multiplication  of  the  tabular  difference  by  the  argu- 
ment difference  is  facilitated  by  certain  little  auxiliary  mul- 
tiplication tables  (called  tables  of  " proportional  parts") 
printed  in  the  margins  of  many  mathematical  tables.  In 
the  example  given  above,  the  tabular  difference  was  16 ;  and 
Table  3  contains  on  the  proper  page  (p.  181)  a  proportional 
part  table  headed  with  this  same  number  16 ;  and  it  shows 
that  for  an  argument  difference  .2,  and  tabular  difference  16, 
the  interpolation  difference  is  3.2,  just  as  we  found  above. 

Other  examples  of  logarithms  are  : 

log         427  =  2.63043,  log    42765  =  4.63109, 

log       4276  =  3.63104,  log  282374  =  5.45082, 

log    0.4276  =  9.63104,  log            2  =  0.30103, 

log  0.42765  =  9.63109,  log     .0027  =  7.43136. 

The  above  considerations  are  preparatory  only  to  the 
actual  use  of  Table  3  ;  and  they  are  not  yet  quite  complete. 
For  it  is  still  necessary  to  explain  the  inverse  use  (p.  12)  of 
the  table,  or,  in  other  words,  the  finding  of  the  number  to 
which  a  given  log  belongs.  Thus,  if  the  given  log  were 
3.42781,  we  should  begin  by  looking  up  its  decimal  part 
among  the  logs  in  the  table.  Finding  it  there,  we  take  out 
the  number  to  which  it  belongs,  2678.  We  then  put  in  the 
decimal  point  according  to  the  whole  number  part  of  the  log. 
This  being  3,  we  know  (p.  24)  that  the  number  required  must 
contain  4  digits.  Therefore  : 

number  to  which  the  log  3.42781  belongs  =  2678. 

1  Except  when  a  glance  at  the  table  shows  that  the  tabular  num- 
bers are  growing  smaller,  in  which  case  the  interpolation  difference 
must  be  subtracted.  This  never  occurs  in  Table  3,  but  happens  fre- 
quently in  Table  4. 


DEAD  RECKONING  WITH  LOGARITHMS  29 

If  the  given  log  had  been  2.42781,  the  table  would  furnish 
the  same  number  2678,  but  the  decimal  point  would  be 
differently  located.  Because  the  whole  number  part  of  the 
given  log  is  now  2,  we  know  that  the  number  to  which  it 
belongs  has  three  digits,  and  so : 

number  to  which  the  log  2.42781  belongs  =  267.8. 

When  the  given  log  is  not  to  be  found  in  the  table  exactly, 
a  process  of  inverse  interpolation  is,  of  course,  necessary. 
Thus,  if  the  given  log  is  4.42784,  we  look  for  its  decimal 
part  in  the  table,  and  find  it  lies  between 

42781,  which  belongs  to  the  number  2678,  and 
42797,  which  belongs  to  the  number  2679. 

The  decimal  part  of  the  given  log  being  42784  is  greater  by 
3  than  the  nearest  tabular  number  42781.  This  3  is  there- 
fore the  interpolation  difference.  The  tabular  difference  is 
16,  obtained  by  subtraction  between  42781  and  42797.  We 
now  divide  the  interpolation  difference  by  the  tabular  dif- 
ference, which  gives  .2  (^  =  0.2,  in  round  numbers).  This 
.2  is  the  argument  difference,  and  therefore  the  complete 
number  belonging  to  the  decimal  part  of  the  log  (42784) 
is  26782.  The  whole  number  part  of  the  given  log 
being  4,  the  required  number  must  have  5  digits,  and  will 
therefore  be  26782.  Had  the  given  log  been  2.42784,  we 
should  have  arrived  at  the  number  26782  in  just  the  same 
way;  but  we  should  locate  the  decimal  point  differently. 
The  whole  number  part  of  the  log  being  now  2,  there  should 
be  only  3  digits  in  the  number,  and  we  should  have : 
number  to  which  the  log  2.42784  belongs  =  267.82. 

Other  similar  examples  are : 

log  =  2.71828,  corresponding  number  =  522.73, 
log  =  4.26323,  corresponding  number  =  18333, 
log  =  9.26323,  corresponding  number  =  0.18333, 
log  =  0.21000,  corresponding  number  =  1.6218. 

The  reader  will  perceive,  from  a  consideration  of  these 
interpolated  numbers,  that  work  with  logarithms  is  never 


30  NAVIGATION 

exact,  absolutely.  This  is  inherent  in  the  nature  of  our 
log  tables,  which  really  contain  only  the  decimal  parts  of  the 
logs  carried  out  to  five  places  of  decimals.  Further  decimals 
of  course  exist,  but  are  here  omitted,  because  five  places 
always  give  sufficient  accuracy  for  navigation  calculations. 

The  simplest  calculations  which  are  facilitated  by  loga- 
rithms are  the  ordinary  arithmetical  processes  of  multi- 
plication and  division.  These  processes  can  be  turned  into 
addition  and  subtraction  by  the  use  of  the  following 
principle : 

The  log  of  a  product  is  equal  to  the  sum  of  the  logs  of  the 
factors. 

According  to  this  principle,  if  we  wish  to  multiply  a  series 
of  factors,  we  simply  add  their  logs.  The  sum  is  then  a  log 
and  the  number  to  which  this  log  belongs  is  the  product  of  the 
series  of  factors.  Suppose,  for  instance,  we  wish  to  multiply 
the  factors  2, 3,  and  4.  The  product  should  be  24.  Proceed- 
ing with  logs,  we  have  from  Table  3 : 

log  2  =  0.30103, 

log  3  =  0.47712, 

log  4  =  0.60206, 

log  product  =  sum  =  1.38021, 

and  the  number  to  which  the  log.  1.38021  belongs  is,  accord- 
ing to  Table  3,  24.00,  the  correct  product. 

It  is  evident  that  the  use  of  the  log  table  is  here  of  no 
advantage,  because  the  factors  are  very  small :  but  when 
large  numbers  are  to  be  multiplied  the  advantage  is  very 
great. 

Taking  now  a  similar  simple  example  of  division,  let  us 
divide  6  by  3.  In  division,  evidently,  we  must  subtract 
the  log  of  the  divisor  from  the  log  of  the  dividend,  to  obtain 
the  log  of  the  quotient.  We  have 

log  6  =  0.77815, 

log  3  =  0.47712, 

log  |  =  difference  =  0.30103, 


DEAD  RECKONING  WITH  LOGARITHMS  31 

and  the  number  to  which  the  log  0.30103  belongs  is  2.000, 
the  correct  quotient.     Other  examples  are: 

2.426  X  42.78  X  17.26  =  1791 .3, 

6.242  X  87.24  X  62.71  =  34149, 
2802 


1.6234, 


^  =  0.75. 


1726 

18 
24 


In  the  last  example,  we  have 

log  18  =  1.25527, 
log  24  =  1.38021. 

The  subtraction  would  lead  to  a  negative  log  because  1.38021 
is  larger  than  1.25527.  Therefore  we  arbitrarily  increase 
1.25527  by  10,  giving  11.25527,  and  then  the  subtraction 

gives 

log  quotient  =  9.87506, 

which  is  the  log  belonging  to  the  number  0.75,  the  correct 
quotient. 

We  come  now  to  the  solution  of  the  two  fundamental 
problems  of  dead  reckoning  (pp..  8,  10)  by  means  of  logs. 
For  this  purpose  we  must  use  our  Table  4,  in  connection  with 
Table  3.     Table  4  is  called  a  trigonometric  log  table  and 
the  tabular  numbers  in  it  are  certain  logs  known  as : 
sine,         abbreviated  sin,     cotangent,  abbreviated  cot, 
cosine,     abbreviated  cos,     secant,        abbreviated  sec, 
tangent,  abbreviated  tan,    cosecant,     abbreviated  esc. 

It  is  not  our  purpose  to  consider  the  theory  of  trigonom- 
etry, but  it  is  necessary  for  the  reader  to  have 
some  understanding  of  its  practical  applica- 
tions. If  we  have  a  triangle  QPY  (fig.  6),  we 
notice  that  it  is  made  up  of  six  "parts,"  the 
three  sides  and  the  three  angles.  Now  it  is  a 
fact  that  if  we  know  any  three  of  these  six  y 


parts,  we  can  calculate  the  other  three  parts,    FIG.  6.-  Trigo- 
provided  one  of  the  known  parts  is  a  side. 
Trigonometry  is  the  branch  of  mathematics  which  enables  us 


32  NAVIGATION 

to  do  this,  and  the  triangle  QPY  is  the  very  triangle  which 
occurs  in  the  two  problems  of  dead  reckoning. 

In  trigonometry,  every  angle  has  belonging  to  it  a  sin, 
cos,  etc.,  just  as  every  number  has  its  log.  These  sines, 
etc.,  can  be  taken  out  of  Table  4  by  means  of  a  pair  of  argu- 
ments in  the  usual  way.  The  two  arguments  are  the  number 
of  degrees  and  the  number  of  minutes  in  the  angle  (p.  9). 
The  number  of  degrees  is  found  in  Table  4  at  the  top  or  bottom 
of  the  page,  and  the  number  of  minutes  in  the  right-hand  or 
left-hand  column.  Each  page  (as,  for  instance,  p.  229)  has 
eight  degree  numbers,  four,  33°,  (213°),  (326°),  and  146°  at 
the  top,  and  four,  123°,  (303°),  (236°),  and  56°  at  the  bottom. 
The  proper  sines,  etc.,  for  all  these  degrees  appear  on  the 
same  page  (p.  229).  When  the  degree  number  is  at  the  top 
or  bottom  of  the  left-hand  column  33°,  (213°),  (303°),  and 
123°,  the  minutes  must  be  taken  from  the  left-hand  column. 
But  when  the  number  of  degrees  is  at  the  top  or  bottom  of  the 
right-hand  column  146°,  (326°),  (236°),  and  56°,  the  minutes 
must  come  from  the  right-hand  column.  And  when  the 
number  of  degrees  comes  »from  the  top  of  the  page,  we  must 
look  for  the  proper  sine,  etc.,  in  a  column  having  the  word 
sin,  etc.,  at  the  top.  But  when  the  degree  number  comes 
from  the  bottom  of  the  page,  the  sine,  etc.,  will  be  taken 
from  a  column  having  the  word  sin,  etc.,  at  the  bottom. 
Thus  (p.  229) : 

sin  33°  26'  =  sin  146°  34'  =  cos  56°  34'  =  cos  123°  26'  =  9.74113. 

In  this  way,  sines,  tangents,  etc.,  can  be  taken  from 
Table  4.  Examples  are : 

sin  28°  32'  =  9.67913,  cot  117°  10'  =  9.71028, 
cos  66°  14' =  9.60532,  sec  12°  40' =  0.01070, 
tan  128°  28'  =  0.09991,  esc  111°  11'  =  0.03038. 

These  sines,  etc.,  are  really  all  logs.  When  the  whole  num- 
ber part  is  9,  it  indicates  that  the  log  belongs  to  a  number 
which  is  wholly  decimal  (see  p.  24),  and  that  the  log  has 
been  arbitrarily  increased  by  10. 


DEAD  RECKONING  WITH   LOGARITHMS  33 

Of  course  these  trigonometric  tables  can  also  be  used  in 
the  inverse  manner.  Thus,  to  find  the  angle  corresponding 
to  the  sin  9.28190,  we  turn  to  p.  207,  and  finding  9.28190  in 
the  sin  column,  we  see  that  the  corresponding  angle  is 
either  11°  2',  191°  2',  168°  58',  or  348°  58'.  When  the  sin, 
etc.,  cannot  be  found  in  the  table  exactly,  we  may  always 
take  the  nearest  one  :  interpolation  is  never  practically 
necessary  in  using  the  trigonometric  tables  in  navigation. 
Examples  are  : 

sec  =  0.17177,  angle  =  47°  40',  227°  40',  132°  20',  or  312°  20', 
tan  =  0.17177,  angle  =  56°  3',  236°  3',  123°  57',  or  303°  57', 
sin  =  9.17177,  angle  =  8°  32',  188°  32',  171°  28',  or  351°  28', 
cos  =  9.17177,  angle  =  81°  28',  261°  28',  98°  32',  or  278°  32', 
esc  =  0.17177,  angle  =  42°  20',  222°  20',  137°  40',  or  317°  40', 
cot  =  0.17177,  angle  =  33°  57',  213°  57',  146°  3',  or  326°  3'. 

Having  thus  explained  the  use  of  Table  4,  we  shall  now 
apply  it  to  the  two  problems  of  dead  reckoning.  These 
problems  are  : 

1.  To  find  latitude  difference  and  departure  from  course 
and  distance  ; 

2.  To  find  course   and  distance  from  latitude  difference 
and  departure. 

These  problems  are  solved  by  means  of  the  following 
formulas,  in  which  the  letter  C  represents  the  course  angle  : 

,j.     f  log  lat.  diff.  =  log  dist.  +  cos  C, 
\  log  dep.         =  log  dist.  +  sin  C. 
tan  C  =  log  dep.  -  log  lat.  diff., 


log  dist.        =  log  dep.  -  sin  C. 

Sometimes  it  is  preferable  to  find  the  distance  from  the 
latitude  difference  instead  of  the  departure.  We  then  use 
the  following  modification  of  formula  (2)  : 

(2')  log  dist.  =  log  lat.  diff.  -  cos  C. 

Let  us  now  solve  with  these  formulas  our  former  problem 
(p.  18),  in  which  a  ship  traveled  1377  miles  on  a  course  of 
166°.  Applying  formula  (1)  above,  we  have: 


34  NAVIGATION 

log  dist.  (1377)  =  3.13893  log  dist.  (1377)  =  3.13893 

cos  C  (166°)  =  9.98690  sin  C  (166°)  =  9.38368 

sum  =  log  lat.  diff.         =  3.12583  *  sum  =  log  dep.  =  2.52261 l 

corresponding  lat.  diff.  =  1336.1  corresponding  dep.  =  333.1 

These  corresponding  latitude  difference  and  departure 
agree  very  closely  with  the  results  already  found  (p.  18) 
from  Table  1. 

If  the  departure  and  latitude  difference  were  given,  we 
could  find  the  course  and  distance  by  means  of  formula  (2)- 
In  the  present  case  we  have  : 

log  dep.  (333.1)  =  2.52261     log  dep.  (333.1)  =  2.52261 

log  lat.  diff.  (1336.1)  =3.12583     sin  C  (166°)  =  9.38368 

by  subtraction,  tan  C  =  9.3967S2  by  subtraction,  log  dist.  =  3.13893 3 
corresponding  C         =  166°          corresponding  dist.          =  1377 

These  numbers,  166°  and  1377  miles,  are  the  same  numbers 
with  which  we  began  this  calculation ;  so  it  is  clear  that  the 
log  method  of  calculation  agrees  with  the  traverse  table 
method.  For  accuracy  the  log  method  is  superior. 

The  transformations  of  departure  into  longitude  differ- 
ence, and  vice  versa,  are  accomplished  logarithmically  with 
the  following  formulas : 

(3)  log  long.  diff.  =  log  dep.—  cos  middle  lat. 

(4)  log  dep.  =  log  long.  diff.  +  cos  middle  lat. 

Thus  the  longitude  difference  corresponding  to  dep.  333.1 
would  be  calculated  by  formula  (3)  as  follows : 

log  dep.  (333.1)  =2.52261 

.cos  mid.  lat.  (29°  16',  p.  18)  =  9.94069 
by  subtraction,  log  long.  diff.  =  2.58192 
corresponding  long.  diff.  =  381'.9  =  6°  21'.9. 

1  These  numbers  have  been  diminished  by  10,  to  allow  for  the  fact 
that  both  cos  C  and  sin  C  have  been  arbitrarily  increased  by  10  (p. 
32;  cf.  also  p.  25). 

2  This  number  has  been  increased  by  10,  and  therefore  is  in  accord 
with  the  usual  practice  of  avoiding  negative  whole  numbers  in  the 
trigonometric  Table  4. 

3  This  subtraction  is  correct,  if  we  remember  that  the  9.38368  is 
really  too  large  by  10. 


DEAD  RECKONING  WITH  LOGARITHMS  35 

This  is  in  close  accord  with  the  result  on  p.  18,  where 
Table  2  gave  6°  20'. 5.  The  logarithmic  method  is  again 
the  more  precise,  for  it  takes  account  of  minutes  in  the  course, 
which  were  neglected  on  p.  18.  But  either  result  is  accurate 
enough  for  practical  purposes. 

Before  finally  leaving  these  problems  of  dead  reckoning, 
we  shall  explain  briefly  two  additional  methods  of  solving 
them  which  differ  from  the  method  so  far  employed.  These 
two  additional  methods  are  called  "Mercator  sailing"  and 
" great  circle  sailing";  whereas,  up  to  the  present,  we  have 
been  using  " middle  latitude  sailing,"  so  named  because 
the  middle  latitude  appears  in  the  calculations. 

Mercator  sailing  is  based  on  a  kind  of  chart  first  designed 
by  Gerhard  Mercator,  a  sixteenth  century  geographer. 
Such  charts  are  still  widely  used  for  nautical  purposes. 
In  calculations  based  on  them,  every  parallel  of  latitude  is 
referred  directly  to  the  equator  by  means  of  a  table  of  "  merid- 
ional parts."  Our  Table  5  is  such  a  table,  and  it  gives  the 
meridional  part  for  every  degree  and  minute  of  latitude 
from  the  equator  to  60°.  These  meridional  parts  are  really 
the  distances  from  the  equator  to  the  several  parallels  of 
latitude,  such  as  they  would  appear  on  a  Mercator  chart 
drawn  to  such  a  scale  that  I'  of  longitude  at  the  equator  would 
occupy  one  linear  unit  on  the  chart.  Thus  the  meridional 
part,  for  lat.  40°  is  given  in  Table  5  as  2607.6.  Suppose  the 
scale  of  the  chart  at  the  equator  were  1  inch  to  the  degree  of 
longitude.  That  would  be  -£$  inch  to  the  minute.  The  dis- 
tance on  the  chart  from  the  equator  to  the  40°  parallel  of 
latitude  would  then  be  2607.6  X  ^V  inches  =  43.46  inches. 
It  is  needless  to  say  that  a  chart  on  such  a  scale  could  not 
show  a  very  large  part  of  the  ocean  on  a  single  sheet. 

Calculations  by  Mercator  sailing  are  of  course  only  made 
when  the  distances  involved  are  large  and  great  accuracy  is 
required.  It  is  therefore  best  to  do  them  by  means  of 
logarithms,  although  it  is  also  possible  to  obtain  Mercator 
results  from  the  traverse  table .  In  such  calculations  we  do  not 


36  NAVIGATION 

use  the  latitude  difference  of  ordinary  middle  latitude  sailing. 
In  its  place  appears  the  " meridional  latitude  difference"  (ab- 
breviated mer.  lat.  diff.),  defined  as  the  difference  between  the 
meridional  parts  (Table  5)  belonging  to  the  two.  latitudes 
(initial  and  final)  involved  in  the  problem.  With  this  defini- 
tion in  mind  we  may  now  give  the  Mercator  formulas  as 
follows : 

(5)  log  mer.  lat.  diff.  =  log  long.  diff.  +  cot  C. 

(6)  log  long.  diff.        =  log  mer.  lat.  diff.  +  tan  C. 

(7)  tan  C  —  log  long.  diff.  —  log  mer.  lat.  diff. 

Let  us  now  apply  these  formulas  to  the  problem  of  pp.  18 
and  33,  in  which  a  ship  starts  from  the  initial  lat.  40°  24'  N. ; 
long.  73°  58'  W.,  and  travels  1377  miles  on  a  course,  C, 
of  166°.  What  final  latitude  and  longitude  does  she  at- 
tain ?  The  latitude  difference  is  found  in  the  ordinary  way 
(p.  34),  there  being  no  special  Mercator  formula  for  it,  and 
comes  out  1336.1  miles,  or  1336M  =  22°  16'.  The  final  lati- 
tude (p.  18)  is  therefore  40°  24'  -  22°  16r  =  18°  8'.  Then, 
from  Table  5,  we  have : 

for  initial  lat?.  40°  24',  mer.  parts  =  2638.9 
for  final  lat.  18°  8',  mer.  parts  =  1099.4 
by  subtraction,1  mer.  lat.  diff.  =  1539.5 

Now,  applying  formula  (6),  we  have : 
log  mer.  lat.  diff.  (1539.5)  (Table  3,  p.  179)  =  3.18738 
tan  C  (166°)  (Table  4,  p.  209)  =  9.39677 

by  addition,  log  long.  diff.  =  2.58415 

corresponding  long.  diff.  (Table  3,  p.  183)   =  383'.8  =  6°  24' 

The  final  longitude  is  therefore  73°  58'  -  6°  24'  =  67°  34'  W., 
whereas  we  obtained  before  67°  38'  W.  (p.  18). 

Finally,  we  shall  apply  the  Mercator  method  to  the 
example  of  p.  21.  It  is  required  to  find  the  course  and 
distance  from 

Sandy  Hook,  lat.  40°  28'  N. ;  long.  73°  50'  W.  to 
St.  Vincent,    lat.  16°  50'  N. ;  long.  25°    7'  W. 

1  If  one  latitude  were  in  the  southern  hemisphere  and  the  other 
in  the  northern,  we  should  add  the  meridional  parts. 


DEAD  RECKONING  WITH   LOGARITHMS  37 

We  have  from  Table  5  : 

for  initial  lat.  40°  28',  mer.  parts  =  2644.2 
for  final  lat.  16°  50',  mer.  parts  =  1018.1 
by  subtraction,  mer.  lat.  diff.  =  1626.1 

The  longitude  difference  is  found  by  subtraction  to  be 
73°  50'  -  25°  7'  =  48°  43'  =  2923'.  Now  applying  formula 
(7),  we  have: 

log  long.  diff.  (2923)  (Table  3)  =  3.46583 
log  mer.  lat.  diff.  (1626)  (Table  3)=  3.21112 
by  subtraction,  tan  C  =  0.25471 

and  therefore  (Table  4)  C  =  119°  5'. 

The  distance  is  found  in  the  ordinary  way  from  the 
latitude  difference  (not  mer.  lat.  diff.)  by  means  of  formula 

(20,  P-  33. 
The  latitude  difference  is  40°  28/-16°  50' =  23°  38' =  1418'. 

Formula  (2')  then  gives  : 

log  lat.  diff.  (1418')  (Table  3)  =  3.15168 
cos  C  (119°  5')  (Table  4)  =  9.686711 

by  subtraction,  log  dist.  =  3.46497 l 

corresponding  dist.  (Table  3)  =  2917 

Course  119°  5',  distance  2917  miles  is  therefore  the 
solution  by  Mercator  sailing.  On  p.  22,  we  obtained  119° 
and  2900  miles;  and  on  p.  19  we  began  with  119°  and  2924 
miles.  The  agreement  is  satisfactory. 

Having  thus  briefly  described  Mercator  sailing,  we  come 
next  to  "great  circle  sailing. "  This  is  a  method  of  determin- 
ing the  ship's  course  toward  her  port  of  destination  in  such  a 
way  that  the  distance  to  be  traveled  will  be  as  short  as 
possible.  If  the  earth's  surface  were  flat  instead  of  spherical, 
the  shortest  course  would  be  a  straight  line,  as  used  in  plane 
sailing;  but  on  the  sphere  the  shortest  course  is  a  curve 
called  a  "  great  circle."  Evidently,  on  all  long  voyages,  the 
great  circle  course  is  the  most  advantageous  one;  that 
mariners  do  not  more  frequently  use  it  is  due  to  a  peculiarity 
of  their  charts. 

1  This  log  is  really  too  large  by  10,  so  the  subtraction  is  correct. 


38  NAVIGATION 

We  cannot  here  enter  into  the  details  of  chart  ''pro- 
jections," as  the  theory  of  chart  making  is  called.  It  is 
sufficient  to  remark  that  a  straight  line  drawn  on  the  ordi- 
nary nautical  charts  (which  follow  the  Mercator  system), 
between  any  two  ports,  will  not  represent  the  shortest  (or 
great  circle)  course  between  them.  On  such  a  chart,  the 
great  circle  course  between  the  two  ports  will  appear  to  be 
longer  than  the  straight  line  course,  although  it  is  really 
shorter.  This  accounts  for  the  use  of  the  longer  Mercator 
course  by  many  navigators. 

Now  there  is  a  kind  of  chart,  called  a  " great  circle  sailing" 
chart,  on  which  straight  lines  between  ports  really  represent 
shortest  (or  great  circle)  courses.  One  would  therefore 
naturally  suppose  that  mariners  would  entirely  discontinue 
the  use  of  Mercator  charts  in  favor  of  great  circle  charts. 
But  there  is  a  reason  for  not  doing  this. 

On  Mercator  charts,  all  terrestrial  longitude  meridians 
are  represented  by  parallel  vertical  straight  lines.  Conse- 
quently, if  we  draw  another  straight  line  on  the  Mercator 
chart  joining  two  ports,  that  line  will  make  the  same  course 
angle  (p.  10)  with  all  the  meridians.  In  this  way,  a  navigator 
can  get  from  a  Mercator  chart,  by  simply  drawing  a  straight 
line,  and  quite  without  calculation,  a  course  angle  which  will 
carry  him  from  one  port  to  another.  And  because  the  course 
angle  so  obtained  is  the  same  with  respect  to  all  meridians 
to  be  crossed  by  the  ship  it  follows  that  the  voyage  can  be 
completed  (theoretically  at  least)  from  the  one  port  to  the 
other  with  the  great  advantage  of  never  changing  the  course 
to  be  steered. 

On  the  other  hand,  the  great  circle  track  makes  a  different 
angle  with  every  meridian  it  passes :  so  that  the  mariner 
must  make  very  frequent  changes  in  the  course  angle  to  be 
steered  during  the  progress  of  a  voyage.  The  simple 
Mercator  track,  without  change  of  course,  is  called  a  "  rhumb 
line"  ;  the  serious  objection  to  it  is  that  it  sometimes  leads 
to  greatly  (and  unnecessarily)  lengthened  voyages. 


DEAD  RECKONING  WITH  LOGARITHMS  39 

The  final  conclusion  is  that  Mercator  charts,  on  account  of 
their  simplicity,  are  most  convenient  for  short  voyages,  or 
for  parts  of  long  voyages  when  the  land  is  not  far  away. 
But  for  shaping  the  main  part  of  the  course  on  a  very  long 
voyage,  great  circle  sailing  charts  are  to  be  preferred. 

At  times,  in  order  to  avoid  very  high  latitudes,  or  to  round 
some  projecting  point  of  land,  navigators  must  substitute  for 
a  single  great  circle  track  one  " composed"  of  two  or  more 
shorter  arcs  of  great  circles.  This  is  called  " composite" 
sailing. 

Finally,  for  the  sake  of  completeness,  we  shall  merely 
mention  two  other  kinds  of  sailing.  " Parallel"  sailing,  which 
is  simply  middle  latitude  sailing  when  the  latitude  difference 
is  zero;  and  " traverse"  sailing,  from  which  the  traverse 
table  gets  its  name.  This  is  also  the  same  thing  as  middle 
latitude  sailing;  but  the  special  word  " traverse"  is  used 
when  the  ship  changes  her  course  frequently,  perhaps  even 
during  a  single  day.  It  is  then  possible  to  sum  up  the 
result  of  all  the  short  courses  which  together  make  up  the 
day's  run.  It  is  merely  necessary  to  take  from  the  traverse 
table  the  latitude  difference  and  departure  for  each  short  course 
separately,  and  then  to  add 1  all  the  values  of  latitude  differ- 
ence for  a  "summed  latitude  difference,"  and  all  the  values 
of  departure  for  a  " summed  departure."  With  these  a 
" composite  course  and  distance"  can  be  taken  from  the 
traverse  table,  or  calculated  with  logs,  and  these  will  repre- 
sent the  motion  of  the  ship,  just  as  if  she  had  steered  an 
unchanged  course  during  the  entire  day. 

1  It  is  necessary  to  sum  separately  latitude  differences  represent- 
ing northward  motion  of  the  ship  and  those  representing  southward 
motion.  The  difference  of  the  two  sums  is  what  we  need  to  know. 
The  same  is  true  of  departures  representing  eastward  and  westward 
motion  of  the  ship. 


CHAPTER  IV 
THE  COMPASS 

THE  ship's  course  has  been  defined  (p.  8)  as  the  angle 
between  the  north  and  the  direction  in  which  the  ship  is 
sailing.  To  ascertain  what  this  angle  is,  or,  in  other  words, 
to  steer  the  ship,  mariners  use  the  compass.  The  dial  (or 
"card")  of  this  instrument  is  divided,  like  any  circle,  into 
360°.  In  the  United  States  Navy  these  are  numbered  in 
such  a  way  (fig.  7)  that  0°  appears  at  the  north,  90°  at  the 
east,  180°  at  the  south,  and  270°  at  the  west.  The  numbers 
therefore  increase  in  a  " clockwise"  direction.  There  are 
also  compasses  in  which  the  numbering  begins  with  0°  at 
both  the  north  and  south  points,  and  increases  to  90°  at  the 
east  and  west  points.'  But  the  United  States  Navy  system 
of  numbering  is  to  be  preferred. 

In  addition  to  the  above  division  and  numbering,  the  dial 
is  also  divided  into  32  points  (pp.  10,  15),  each  containing 

oar\o 

^-,  or  11|°.     These  points  are  then  further  subdivided 
32 

into  quarter  points,  all  of  which  is  shown  clearly  in  Fig.  7. 

The  naming  of  the  points  has  not  been  done  by  chance, 
but  in  accordance  with  a  definite  rule.  The  four  principal, 
or  "cardinal,"  points  are  north,  east,  south,  and  west.  The 
remaining  points  are  located  by  a  continued  process  of 
halving.  Halfway  between  the  cardinal  points  are  the 
"inter-cardinal"  points;  and  each  is  named  by  combining 
the  names  of  the  two  cardinal  points  adjacent  to  it.  Thus 
northeast  (abbreviated  N.E.)  is  halfway  between  north 
and  east.  Again  halving  and  combining  names,  we  get 
points  like  E.N.E.,  S.S.E.,  etc.  Still  once  more  halving 
completes  the  tally  of  32  points :  but  a  combination  of 
names  would  now  be  too  complicated.  However,  since 

40 


THE   COMPASS 


41 


each  of  these  final  points  must  necessarily  be  adjacent  to  a 
cardinal  or  inter-cardinal  point,  they  are  named  by  simply 
increasing  the  name  of  such  adjacent  cardinal  or  inter- 
cardinal  point.  This  is  accomplished  with  the  word  "by." 


FIG.  7.  —  Compass  Card. 

Thus  we  find,  adjacent  to  N.E.,  the  points  N.E.byE.,  and 
N.E.  by  N.  In  the  light  of  the  above,  it  is  easy  to  "box" 
the  compass,  as  seamen  say,  or  to  name  the  32  points  in 
order. 

When  the  point  system  of  division  is  used,  and  an  accuracy 


42  NAVIGATION 

closer  than  a  single  point  is  required,  the  compass  card  is 
still  further  subdivided  into  quarter  points.  In  naming 
these  it  is  customary,  in  the  United  States  Navy,  to  "box" 
from  N.  and  S.  towards  E.  and  W.  Thus  the  space  between 
N.N.E.  and  N.E.  byN.  would  be  divided  into  four  parts 
thus:  N.N.E.JE.,  NrN.E.JE.,  N.N.E.fE.  But  an  excep- 
tion is  made  to  this  last  rule  in  the  case  of  quarter  points 
adjacent  to  a  cardinal  or  inter-cardinal  point.  These  last 
are  always  put  first  in  naming  the  quarter  points.  Thus, 
between  E.  by  N.  and  E.,  if  we  always  boxed  from  N.  towards 
E.,  we  should  have :  E.  by  N.JE.,  E.  by  N.|E.,  E.  by  N.fE. 
But  it  is  customary,  because  shorter,  to  name  these  quarter 
points  E.fN.,  E.JN.,  and  E.JN. 

Inside  the  "bowl"  of  the  compass,  and  adjacent  to  the 
card,  a  black  line  is  marked  on  the  bowl.  This  line  is  in 
plain  view  of  the  steersman,  through  the  glass  cover  of  the 
compass,  and  is  called  the  "lubber  line."  When  the  ship 
is  headed  in  such  a  way  that  this  line  comes  opposite  N.E., 
for  instance,  on  the  card,  the  ship  will  be  on  a  N.E.  course, 
which  makes  an  angle  of  45°  with  the  north. 

But  would  the  ship  really  be  traveling  on  a  line  making 
a  45°  angle  with  the  geographic  meridian,  or  direction  of 
the  north  pole  of  the  earth?  She  would  be  doing  so  only 
if  the  compass  were  absolutely  correct.  This  is  practically 
the  case  with  the  "gyro-compass,"  a  mechanical  contrivance 
now  much  used  in  the  navy,  but  not  the  case  with  the  ordi-' 
nary  "magnetic"  compass. 

In  Chapters  II  and  III,  concerning  dead  reckoning,  we  have 
always  used  the  word  "course"  as  if  all  compasses  were 
absolutely  correct.  But  since  they  are  not  correct,  it  is 
now  necessary  to  make  allowance  for  their  errors.  In  other 
words,  whenever  we  use  a  compass,  we  must  first  ascertain 
the  difference  between  the  "true  course"  and  the  "compass 
course."  It  must  not  be  supposed  from  this  statement  that 
a  ship  can  be  steered  on  two  different  courses  at  the  same 
moment.  There  is  really  only  one  direction  along  which 


THE   COMPASS  43 

the  ship  is  moving:  but  the  angle  between  that  direction 
and  the  true  north  may  be  different  from  the  angle  between 
it  and  the  "  compass  north."  It  is  the  course  measured 
from  the  true  north  that  must  be  used  in  all  dead-reckoning 
calculations,  and  that  always  results  from  such  calculations : 
but  for  steering  the  ship  by  means  of  a  compass  the  steers- 
man must  be  furnished  with  the  course  as  measured  from 
the  compass  north.  Therefore  it  is  essential  for  the  navigator 
to  know  the  difference  between  the  two.  This  difference 
is  called  the  " error"  of  the  compass. 

Unfortunately,  this  error  is  made  up  of  two  parts.  The 
first,  called  " variation"  of  the  compass,  is  due  to  peculiari- 
ties in  the  earth's  magnetism,  and  is  quite  different  in  dif- 
ferent places  on  the  earth.  It  also  varies  in  different  years 
at  the  same  place.  But  at  any  one  time,  all  ships  in  the 
same  part  of  the  ocean  will  have  the  same  variation. 

The  mariner  can  always  ascertain  how  great  the  varia- 
tion is  in  his  part  of  the  ocean,  because  it  is  always  marked 
on  his  chart.  Certain  curved  lines  are  drawn  on  the  chart ; 
and  if  the  ship  is  located  on  or  near  a  line  marked  "  varia- 
tion 10°,"  for  instance,  it  follows  that  the  navigator  must 
on  that  day  allow  for  10°  of  variation.  It  is  also  important 
to  take  into  consideration  possible  changes  in  the  variation. 
Sometimes  the  annual  change  is  marked  on  the  chart;  if 
not,  it  is  important  to  use  a  chart  of  recent  date. 

The  second  part  of  the  error  is  called  " deviation"  and  is 
due  to  peculiarities  in  the  magnetism  always  developed  in 
the  metallic  parts  of  the  ship  itself.  It  is  different  in  dif- 
ferent ships,  even  in  the  same  part  of  the  ocean,  and  is  even 
different  in  the  same  ship,  when  she  is  headed  on  different 
courses.  Methods  have  been  invented  for  "  compensating " 
marine  compasses,  so  as  to  remove  the  effects  of  deviation, 
and  these  methods  are  quite  effective.  But  even  when 
they  are  used,  it  is  necessary,  before  beginning  a  long  voyage, 
to  have  a  " compass  adjuster"  visit  the  ship.  He  will  then 
"swing"  the  ship  on  a  number  of  different  courses,  and 


44  NAVIGATION 

adjust  the  compass  so  that  it  will  be  as  nearly  correct  as  pos- 
sible. Finally,  he  will  determine,  by  means  of  astronomic  or 
other  observations,  just  what  the  remaining  compass  devia- 
tion is  on  all  the  various  courses,  and  give  the  navigator  a 
table  of  these  remaining  deviations.  This  table  must  be  taken 
into  account  in  "shaping"  the  ship's  course  during  the 
voyage.  The  navigator  must  also,  from  time  to  time,  check 
these  tabular  deviations  while  at  sea  by  means  of  astronomic 
observations  of  his  own,  to  take  care  of  possible  changes. 

Such  astronomic  observations  are  made  with  an  instru- 
ment (the  " azimuth  circle"),  which  can  be  attached  to  the 
compass,  and  with  which  the  " compass  bearing"  of  the 
sun  or  any  other  object  can  be  observed.  The  compass 
bearing  is  simply  the  compass  direction  of  the  object,  as 
seen  from  the  ship ;  or  the  compass  course  on  which  the  ship 
would  be  steered,  if  she  were  moving  directly  toward  the 
object.  When  the. sun  is  used,  its  true  bearing,  measured 
from  the  true  north,  can  be  taken  from  astronomic  tables 
which  will  be  explained  later;  and  it  is  called  the  sun's 
" azimuth."  A  comparison  of  this  true  bearing  with  that 
measured  on  the  compass  with  the  azimuth  circle  then  makes 
the  compass  error  known. 

When  it  is  not  convenient  to  observe  the  sun,  it  is  possible  to 
substitute  observations  of  a  distant  well-defined  terrestrial  ob- 
ject, whose  true  bearing  can  be  measured  on  a  chart  for  com- 
parison with  various  compass  bearings  observed  while  the  ship 
is  being  swung.  Another  method  is  to  set  up  a  compass  on 
shore,  away  from  any  iron  or  steel,  and  use  it  to  determine 
the  bearing  of  the  distant  object.  And  there  is  still  another 
method,  if  the  above  compass  and  the  ship's  compass  are  inter- 
visible.  For  the  bearing  of  each  may  then  be  taken  from  the 
other,  and  these  should  differ  by  exactly  180°.  If  they  do  not, 
the  variation  from  180°  must  be  due  to  deviation  on  board. 

The  "pelorus"  is  another  instrument  which  may  at  times 
replace  the  azimuth  circle.  It  is  located  anywhere  on  the 
ship,  at  a  convenient  point  for  observation,  and  not  neces- 


THE   COMPASS 


45 


sarily  close  to  the  compass.  It  has  a  "dummy  card"  and  a 
lubber  line.  The  dummy  card  can  be  turned  until  the 
lubber  line  indicates  the  same  course  as  the  real  compass. 
Observations  of  bearings  with  the  pelorus  will  then  obviously 
be  the  same  as  if  made  on  the  compass  with  the  azimuth  circle. 
The  advantage  of  the  pelorus  is  that  it  can  be  used  anywhere 
on  board,  while  the  compass  must  be  kept  constantly  in  the 
exact  place  where  it  was  "adjusted"  before  leaving  port. 

The  error  thus  determined  astronomically  or  otherwise 
is  the  sum  of  the  variation  and  deviation.  If  we  indicate 
by  E  the  total  compass  error  in  that  place,  at  that  time,  on 
that  ship,  and  on  that  course ;  by  D  the  deviation  similarly 
described  ;  by  V  the  variation  at  that  time  and  in  that  place ; 
and  if  all  three  are  counted  from  0°  in  the  usual  direction 
around  the  compass  card,  then 
we  have  the  formula  : 

(1)         E  =  V  +  D. 

By  counting  in  the  usual  direc- 
tion, we  mean  counting  from  the 
north  around  to  the  east,  as  all 
courses  are  counted  (p.  19) ;  so 
that  a  compass  error  of  10°,  for 
instance,  would  mean  that  the 
compass  north  pointed  10°  east 
of  the  true  north,  or  had  a  true 
bearing  of  N.  10°  E.  (p.  19). 
This  is  shown  in  Fig.  8,  which 
also  shows  the  ship's  course, 
counted  in  the  same  way. 

It  is  clear  from  the  figure  that  if  we  now  indicate  : 

by  C,  the  ship's  compass  course, 

by  T,  the  ship's  true  course, 

by  E,  the  compass  error, 

we  shall  have  the  formula  : 

(2)       .T  = 


FIG.  8.  —  Compass  Error. 


46  NAVIGATION 

The  simple  formulas  (1)  and  (2)  enable  the  navigator  to 
make  all  necessary  compass  calculations.  The  following 
are  examples. 

Suppose,  for  instance,  that  the  error  E  has  been  deter- 
mined by  observation,  and  the  variation  V  taken  from  the 
chart.  Formula  (I)  then  makes  it  possible  to  calculate 
the  deviation  D.  For  the  formula  shows  that  E  is  the  sum 
of  V  and  D ;  and  so  D  must  be  the  difference  of  E  and  7, 

or:  D=E-V. 

Thus  the  deviation  D  becomes  known,  as  a  check  on  the 
compass  adjuster's  work,  and,  while  this  value  of  D  is  cor- 
rect only  for  the  particular  course  on  which  the  ship  was 
headed  at  the  time  the  observation  was  made,  yet  that 
course  is  the  very  one  for  which  it  is  especially  important 
to  have  correct  information. 

Again,  suppose  dead-reckoning  calculations  show  that  the 
ship  is  to  sail  on  a  40°  course.  These  calculations  always 
furnish  the  true  course  (p.  43)  so  that  T  =  40°.  The 
variation  being  known  from  the  chart,  and  the  deviation 
from  the  adjuster's  table,  we  know  from  (1)  E  =  V  +  D. 
Then  from  (2)  we  see  that  C  =  T  —  E}  which  gives  the 
compass  course.  Let  us  suppose  in  the  present  case,  that 
V  was  9°,  D  1° ;  then  E  =  V  +  D  =  9°  +  1°  =  10° ;  and 
since  T  =  40°,  C  =  T  -  E  =  40°  -  10°  =  30° ;  and  the 
helmsman  would  be  directed  to  steer  a  30°  course  by  com- 
pass. 

If,  in  Fig.  8,  the  compass  north  happened  to  be  10°  on  the 
left  side  of  the  true  north,  instead  of  the  right,  the  error  E 
would  be  350°,  instead  of  10°  (see  also  fig.  7,  p.  40).  This 
might  be  made  up  of  a  variation  V  of  349°  and  a  deviation 
D  of  1°,  as  before.  If  the  true  course  is  again  to  be  40°, 
the  compass  course  would  be  40°  —  350°,  according  to  the 
formula  C  =  T  —  E.  This  subtraction  being  impossible, 
we  increase  the  40°  by  a  complete  circumference  of  360°, 
which  is  always  permissible,  and  then  have : 


THE  COMPASS  47 

C  =  360°  +  40°  -  350°  =  50°. 

The  ship  would  be  steered  on  a  compass  course  of  50°. 

An  alternative  way  to  take  care  of  errors,  variations, 
and  deviations  on  the  left  side  of  the  true  north  is  to  mark 
them  with  the  negative  or  minus  sign.  Instead  of  calling 
V  349°,  we  might  call  it  -  11°.  This  is  really  the  best  way, 
and  leads  to  the  same  result  as  before,  if  we  remember  that 
the  subtraction  of  a  minus  quantity  is  always  equivalent  to 
an  addition.  In  the  example  just  given,  calling  V  —  11°, 
instead  of  349°,  we  should  have :  E=V  +  D=-U°  + 
1°  =  -  10°;  and  C  =  T  -  E  =  40°  -  (-  10°)  =  50°,  the 
same  compass  course  as  before. 

An  older  way  of  designating  variations,  deviations,  and  errors 
is  to  call  them  east  when  the  compass  north  points  to  the 
right  of  the  true  north,  and  west  when  it  points  to  the  left 
of  the  true  north.  This  method  leads  to  the  necessity  of 
providing  various  rules  or  diagrams  with  which  to  make 
compass  calculations.  We  think  the  best  way  to  avoid 
error  (and  such  errors  may  lose  ships  and  lives)  is  to  use  the 
method  here  given  with  its  two  simple  formulas.  When 
some  other  designation  of  the  error,  or  some  other  method 
of  numbering  the  card,  is  demanded  by  a  captain,  it  is  always 
possible  to  conform  to  that  demand,  but  also  to  translate 
every  problem  into  our  method  (in  imagination  at  least) 
as  a  check  against  mistake. 

The  following  is  an  example  of  a  compass  adjuster's  ''devia- 
tion table,"  taken  from  Bowditch's  "Navigator"  (1916 
edition).  The  deviations  are  set  down  in  degrees  and  tenths 
of  a  degree,  instead  of  degrees  and  minutes,  for  convenience 
in  the  further  calculations.  The  ship  was  swung  so  that 
her  head  bore  successively  around  the  horizon,  and  obser- 
vations were  made  at  intervals  of  15°.  This  is  a  smaller 
interval  than  is  usually  necessary ;  and  the  deviations  in  the 
table  are  much  larger  than  commonly  occur  in  a  modern 
well-compensated  compass. 


48 


NAVIGATION 


DEVIATION   TABLE 


BEARING 

BEARING 

BEARING 

BEARING 

OF  SHIP'S 

DEVIA- 

OP SHIP'S 

DEVIA- 

>F SHIP'S 

DEVIA- 

OF SHIP'S 

DEVIA- 

HEAD  BY 

TION 

HEAD    BY 

TION 

rlEAD     BY 

TION 

HEAD    BY 

TION 

COMPASS 

COMPASS 

COMPASS 

COMPASS 

o 

o 

o 

o 

0 

0 

o 

0 

0 

-  15.5 

90 

-  9.1 

180 

+  17.9 

270 

+    9.9 

15 

-  14.9 

105 

-  9.0 

195 

+  23.8 

285 

+    1.9 

30 

-  13.3 

120 

-  7.8 

210 

+  27.1 

300 

-    4.2 

45 

-  11.3 

135 

-  5.9 

225 

+  25.6 

315 

-  10.3 

60 

-  10.0 

150 

-2.3 

240 

+  22.0 

330 

-  13.6 

75 

-    9.7 

165 

+  8.5 

255 

+  15.9 

345 

-  16.0 

To  illustrate  the  use  of  this  table,  let  us  suppose  the  ship 
to  be  sailing  on  a  compass  course  of  165°,  in  a  part  of  the 
ocean  where  the  variation  is  +  10°,  or  10°  E.  Using  formula 
(1)  (p.  45),  and  finding  from  our  table  that  the  deviation  D 
for  165°  is  +  8°. 5,  we  have  the  compass  error  E  =  V  +  D 
=  +  10°  +8°.5  =  +  18°.5.  By  formula  (2)  (p.  45)  the  true 
course  of  the  ship  is  T  =  C  +  E=  165°  +.18°.5  =  183°.5. 
We  should  use  this  true  course  183°. 5  in  calculating  later 
the  ship's  position  by  dead  reckoning  (p.  10). 

If  the  compass  variation  were  everywhere  the  same,  it 
would  be  more  convenient  to  have  a  table  of  compass  errors, 
instead  of  a  table  of  deviations ;  but  because  the  variation, 
as  given  on  the  chart,  varies  greatly,  the  table  must  be 
specially  made  for  deviations  only. 

Equally  important  with  the  above  use  of  our  deviation 
table  is  its  inverse  use.  When  the  navigator  has  calculated 
by  dead  reckoning  the  course  he  must  steer,  that  course, 
as  it  comes  from  the  calculations,  will  be  a  true  course  (p. 
43) ;  and  it  is  necessary  to  turn  it  into  a  compass  course  for 
the  use  of  the  steersman. 

To  do  this  we  must  know  the  deviation ;  and  we  cannot 
get  it  directly  from  the  deviation  table  above,  because  the 
use  of  that  table  presupposes  a  knowledge  of  the  compass 
course,  the  very  thing  we  are  trying  to  find.  The  best 


THE   COMPASS 


49 


way  to  avoid  this  difficulty  is  to  imagine  the  deviation  to  be 
non-existent,  for  the  moment,  and  to  make  use  of  the  "mag- 
netic course,"  defined  as  the  course  which  would  be  indi- 
cated by  the  compass,  if  deviation  were  thus  totally  absent. 
Under  these  circumstances,  formula  (1)  gives  E  =  V,  since 
D  =  0 ;  and  if  we  designate  the  magnetic  course  by  Mt  we 
may  write,  in  place  of  formula  (2)  (p.  45) : 

(3)         M  =  T  -  V. 

Let  us  suppose  a  case  in  which  the  variation  is  +  10°>  and 
the  desired  true  course  of  the  ship  175°.  Then  the  magnetic 
course,  allowing  for  variation  only,  will  be,  by  formula  (3) : 

M  =  T  -  V  =  175°  -  10°  =  165°. 

This  course  is  not  really  a  compass  course,  because  no 
account  has  yet  been  taken  of  the  deviation.  Nor  can  we 
yet  find  the  deviation  directly  from  the  deviation  table, 
because  in  that  table  we  must  still  know  the  compass  course 
to  use  as  the  argument  (p.  10),  whereas  we  know  as  yet  only 
the  magnetic  course.  Therefore  navigators  should  always 
request  the  compass  adjuster  to  furnish  a  " second  deviation 
table,"  in  which  the  argument  is  the  magnetic  course,  in- 
stead of  the  compass  course.  Such  a  second  table  can  al- 
ways be  calculated  from  the  other.  We  here  give  one  that 
has  been  calculated  from  the  table  on  the  preceding  page. 

SECOND   DEVIATION   TABLE 


MAG- 

MAG- 

MAG- 

MAG- 

NETIC 

NETIC 

NETIC 

NETIC 

BEARING 

DEVIA- 

BEARING 

DEVIA- 

BEARING 

DEVIA- 

BEARING 

DEVIA- 

OF SHIP'S 

TION 

OF  SHIP'S 

TION 

OF  SHIP'S 

TION 

OF  SHIP'S 

TION 

HEAD 

HEAD 

HEAD 

HEAD 

o 

o 

o 

o 

o 

0 

o 

0 

0 

-  14.9 

90 

-9.0 

180 

+  11.0 

270 

+  16.5 

15 

-  13.4 

105 

-8.4 

195 

+  16.9 

'285 

+    4.1 

30 

-  11.7 

120 

-  6.9 

210 

+  21.3 

300 

-    7.1 

45 

-  10.4 

135 

'  -4.8 

225 

+  24.9 

315 

-  13.2 

60 

-    9.8 

'150 

-  1.4 

240 

+  26.8 

330 

-  15.7 

75 

-    9.3 

165 

+  5.0 

255 

+  24.1 

345 

-  15.5 

50  NAVIGATION 

We  also  add  as  an  example  the  calculation  of  one  number 
in  the  second  table  from  those  given  in  the  first.  We  shall 
find  the  deviation  corresponding  to  the  magnetic  course 
165°  ;  and  we  do  it  by  a  kind  of  interpolation  (p.  12).  From 
the  first  table  we  have  the  deviation  —  2°.3  for  the  compass 
course  150°.  Since  the  deviation  is  the  only  difference 
between  compass  and  magnetic  courses,  it  follows  that 
150°  -  2°.3,  or  147°.7  magnetic,  corresponds  to  150°  by  com- 
pass. Similarly,  173°.5  magnetic  corresponds  to  165°  by 
compass. 

The  magnetic  course  165°  for  which  we  are  making  the 
calculation  falls  between  147°.  7  and  173°.  5,  and  exceeds 
the  smaller  of  the  two  by  17°.3.  The  whole  difference  be- 
tween 147°.7  and  173°.5  is  25°.8.  Similarly,  the  whole  dif- 
ference between  the  two  compass  courses  involved  is  15°. 
Therefore  we  may  write  the  proportion  : 

25°.8  :  15°  =  17°.3  :  x°, 

where  x  is  the  excess  over  150°  of  the  compass  course  corre- 
sponding to  165°  magnetic. 

Solving  this  proportion  by  the  ordinary  rules  of  arithmetic, 
we  have  : 


The  compass  course  belonging  to  165°  magnetic  is  there- 
fore 150°  +  10°.0  =  160°.0.  The  corresponding  deviation 
is  165°  -  160°.0  =  +  5°.(V  which  is  therefore  the  deviation 
for  165°  magnetic,  and  appears  as  such  in  the  second  table. 
This  entire  table  can  be  computed  from  the  first  table  in  an 
hour. 

Sometimes  the  second  deviation  table  gives  compass  courses 
instead  of  deviations.  It  is  then  often  called  a  "  table  of 

1  A  comparison  of  formulas  (1),  (2),  and  (3)  shows  that 
D  =  M  —  C  ;  so  that  the  deviation  is  obtained  by  subtracting  the 
compass  course  from  the  magnetic  course.  This  is  also  evident 
from  the  definition  of  a  magnetic  course  (p.  49). 


THE   COMPASS  51 

steering  courses " ;  and  in  the  example  just  calculated  it 
would  give  the  compass  or  steering  course  160°  for  the  mag- 
netic course  165°,  instead  of  giving  the  deviation  -f  5°. 

We  shall  still  further  illustrate  this  important  matter  by 
an  example,  supposed  to  occur  on  board  a  ship  for  which 
our  two  deviation  tables  hold  good. 

What  is  the  compass  course  to  be  given  the  helmsman  at 
Sandy  Hook,  on  a  voyage  to  St.  Vincent? 

We  have  already  found,  from  dead-reckoning  calculations 
(p.  22)  the  course  119°.  Being  the  result  of  a  dead-reckon- 
ing calculation,  this  is  a  true  course.  The  track  chart  of 
the  north  Atlantic  gives  the  variation  at  Sandy  Hook  as 
10°  W.,  or  -  10°.  The  true  course  being  119°,  we  get  the 
magnetic  course,  allowing  for  variation  only,  by  formula  (3), 
M  =  77-F  =  119°-(-  10°)  =  129°.  The  second  devia- 
tion table  shows  that : 

for  magnetic  course  120°,  the  deviation  is  —  6°. 9,  and 
for  magnetic  course  135°,  the  deviation  is  —  4°. 8. 

Magnetic  course  129°  falls  between  120°  and  135°,  so  that 
an  interpolation  (to  be  extremely  exact)  between  —  6°. 9 
and  —  4°.8  makes  the  deviation  for  magnetic  course  129° 
come  out  —  5°.6.  Formulas  (1)  and  (2)  now  give  : 

Error  =E  =  V+D=-W°-  5°.6  =  -  15°.6 

Compass  course  =  C  =  T-E  =    119°  -(-  15°.6)  =  134°.6. 

To  check  this,  we  can  now  solve  the  same  problem  in  the 
inverse  way  with  the  first  deviation  table.  For  the  compass 
course  134°. 6,  this  table  gives  the  deviation  as  —  5°. 9.  The 
variation  being  —  10°,  we  have : 

E  =  V  +  D  =  -10°  -  5°.9  =  -  15°.9  and 
T  =  C  +  E  =  134°.6  -  15°.9  =  118°.7, 

agreeing  very  closely  with  the  true  course  119°,  with  which 
we  started.  This  shows  that  the  two  deviation  tables  are 
quite  consistent  in  this  case,  and  also  checks  the  accuracy 
of  the  calculation. 


52 


NAVIGATION 


We  shall  close  this  chapter  with  the  following  little  table, 
showing  the  correspondence  between  the  two  methods  of 
dividing  the  compass  card  into  points,  and  into  degrees. 

COMPASS  POINTS  AND  DEGREES 


°   / 

o    , 

°   / 

°     / 

North 

0  0 

East 

90  0 

South 

180  0 

West 

270  0 

N.  by  E. 

11  15 

E.  by  S. 

101  15 

S.  by  W. 

191  15 

W.  by  N. 

281  15 

N.N.E. 

22  30 

E.S.E. 

112  30 

s.s.w. 

202  30 

W.N.W. 

29230 

N.E.  by  N. 

3345 

S.E.  by  E. 

123  45 

S.W.  by  S. 

213  45 

N.W.  by  W. 

30345 

N.E. 

45  0 

S.E. 

135  0 

s.w. 

225  0 

N.W. 

315  0 

N.E.  by  E. 

56  15 

S.E.  by  S. 

146  15 

S.W.  by  W. 

236  15 

N.W.  by  N. 

326  15 

E.N.E. 

67  30 

S.S.E. 

15730 

W.S.W. 

247  30 

N.N.W.  • 

33730 

E.  by  N. 

7845 

S.  by  E. 

16845 

W.  by  S. 

25845 

N.  by  W. 

34845 

J  pt.   =  2°  49' 


pt.   =  5°  38' 


8°  26' 


1  pt.   =  11°  15' 


CHAPTER  V 
COASTWISE  NAVIGATION 

BEFORE  proceeding  to  a  consideration  of  navigation  by 
means  of  astronomic  observations,  as  it  is  practiced  on  the 
high  seas,  we  must  first  explain  certain  methods  by  which 
it  is  possible  to  ascertain  a  ship's  position  in  latitude  and 
longitude  while  she  is  in  sight  of  land.  Often  such  methods 
suffice  to  complete  a  long  coastwise  voyage  in  safety;  they 
are  always  important  for  a  last  determination  of  the  ship's 
position  before  a  deep-sea  voyage  actually  begins.  Such  a 
last  determination  is  called  " taking  a  departure"  (cf.  p.  2), 
and  from  such  point  of  departure  dead-reckoning  calcula- 
tions begin  for  the  first  day  of  the  voyage. 

Any  determination  or  fixing  of  a  ship's  position,  by  astro- 
nomic observations  or  otherwise,  is  often  called,  for  brevity, 
a  "fix."  To  obtain  one  while  in  sight  of  land  it  is  customary 
to  make  observations  upon  well-known  objects  ashore, 
such,  for  instance,  as  lighthouses,  or  other  conspicuous 
objects  marked  on  the  chart.  It  is  always  possible  to  ob- 
serve the  bearings  of  such  objects  from  the  ship's  deck  with 
the  compass,  azimuth  circle,  or  pelorus  (p.  44). 

When  there  is  but  one  such  object  in  sight,  it  is  impossible 
to  secure  a  fix  with  ordinary  instruments,  if  the  vessel  is 
at  anchor.  But  if  she  is  running,  it  is  merely  necessary  to 
take  two  bearings,  and  to  estimate  the  distance  run  by  the 
ship  in  the  interval  between  the  two.  Figure  9  will  make 
this  matter  clear.  A  lighthouse  ashore  is  at  L.  SS"  is  the 
direction  of  the  ship's  course;  S  her  position  when  the 
first  bearing  was  observed,  and  S'  her  position  at  the  time 
of  the  second  bearing.  SN  is  the  direction  of  the  north. 

53 


54 


NAVIGATION 


After  taking  the  first  bearing,  the  navigator  must  calculate 
the  angle  S"SL,  between  the  ship's  course  SS"  and  the 

lighthouse  direction  SL.  Thus, 
if  the  ship's  course  angle  NSS" 
(p.  10)  was  20°,  and  the  bearing 
NSL  was  42°,  the  angle  S"SL 
would  be  42°  -  20°  =22°.  As 
the  ship  proceeds  on  her  course, 
the  angle  S"SL  will  become 
larger,  and  a  second  bearing  must 
be  taken  at  the  moment  when 
the  ship  reaches  the  point  S', 
where  the  angle  S"SL  has  become 
S"S'L.  This  point  S'  must  be 
so  chosen  that  the  angle  S"S'L 
is  just  twice  the  angle  S"SL  ob- 
served at  S ;  or,  in  this  case,  44°. 
This  is  called  "  doubling  the  bear- 

—  Ship's  Position  by  Two  ing  from  the  bow,"  and  it  can 
easily  be  accomplished  if  we  con- 
tinue watching  the  compass  bearing  of  L  as  the  ship  goes 
ahead,  and  catch  the  observation  at  the  right  moment.  The 
ship's  course  not  having  been  changed  from  20°  (this  is 
important),  the  right  moment  will  occur  when  L  bears 
20°  +  44°  =  64°  by  the  compass. 

It  can  easily  be  proved  by  geometry  that  the  distance 
S'L  between  the  ship  at  S'  and  the  lighthouse  at  L  will  be 
equal  to  the  distance  SS'  traveled  by  the  ship  in  the  inter- 
val between  the  two  observations.  *  This  distance  can  be 
estimated  quite  accurately  with  an  instrument  called  a 
"log,"  or  "patent  log,"  which  is  towed  astern  of  the  ship. 
It  is  so  constructed  that  it  turns  as  it  is  pulled  through  the 
water,  and  the  number  of  turns  is  automatically  counted  by 
an  attached  contrivance  on  deck.  The  count  is  (also  auto- 
matically) turned  into  miles  of  distance ;  so  that  the  log  on 
deck  will  indicate  how  far  the  ship  traveled  from  S  to  S'. 


COASTWISE  NAVIGATION  55 

As  soon  as  we  know  the  distance  S'L  and  the  bearing  of 
the  line  S'L,  we  can  ''lay  down"  or  "plot"  the  position  of 
S'  on  the  chart;  and  this  will  be  a  "good  fix."  To  do  this, 
let  us  indicate  by  B'  the  bearing  of  the  line  S'L,  and  then 
draw  on  the  chart,  through  the  lighthouse  L,  a  pencil  line 
whose  bearing  from  L  is  E'  +  180°,  or  " B'  reversed."  This 
can  be  done  with  a  "course  protractor,"  or  with  "parallel 
rulers,"  instruments  to  be  purchased  from  any  dealer  in 
navigators'  supplies.  Next  we  measure  or  "lay  off"  on  that 
line  the  distance  S'L,  equal  to  the  run  SS'  as  it  came  from 
the  log.  We  always  know  the  right  "scale"  of  the  chart 
(or  fraction  of  an  inch  corresponding  to  one  logged  mile) 
which  must  be  used  in  laying  off  the  distance  S'L;  for  we 
know  that  one  mile  always  corresponds  to  1  minute  of 
latitude  (p.  15),  and  the  right-  and  left-hand  edges  of  the 
chart  are  always  divided  into  degrees  and  minutes  of  latitude. 

Since  the  above  bearings  were  observed  by  compass,  it 
is  now  important  to  consider  the  compass  error  (p.  43). 
This  will  not  affect  the  observations,  because  it  will  be  the 
same  for  both  ship's  course  and  lighthouse  bearing,  so  the 
angles  Sf/SL  and  S"S'L,  which  are  obtained  by  subtraction, 
will  be  the  same  as  if  there  were  no  compass  error.  But 
when  we  come  to  plotting  on  the  chart,  the  compass  bearing 
B'  must  be  corrected  by  adding  the  deviation  from  the 
deviation  table  (pp.  48,  49).  The  resulting  magnetic  bear- 
ing (p.  49)  must  be  used  for  Bf,  if  the  chart  has  printed 
on  it  a  compass  card  (p.  41)  showing  magnetic  bearings. 
If  the  printed  card  shows  true  bearings  only,  B'  must  be 
corrected  for  both  deviation  and  variation  (p.  43). 

A  specially  important  case  of  the  foregoing  occurs  when 
the  two  angles  S"SL  and  S"S'L  are  45°  and  90°.  The 
second  bearing  B'  will  then  put  the  light  just  abeam,  and 
the  distance  by-  log,  SS',  is  the  distance  at  which  the  ship 
passes  the  light  abeam.  This  case  is  called  a  "bow-and- 
beam  bearing."  The  navigator  sights  the  light  when  it  bears 
45°  or  4  points  (p.  52)  "broad"  on  the  bow,  "starboard," 


56  NAVIGATION 

or  "port."  He  then  "reads"  the  log.  When  he  brings 
the  light  abeam  through  the  motion  of  the  ship,  he  reads 
the  log  again,  and  the  run  in  the  interval,  as  taken  from  the 
log,  is  the  light's  distance  abeam. 

When  sailing  along  the  coast,  it  is  particularly  important 
so  to  shape  the  ship's  course  that  lights  and  other  promi- 
nent landmarks  will  be  passed  at  the  right  distance  abeam. 
The  chart  shows  what  the  right  distance  is  :  if  the  navigator 
shapes  a  course  which  makes  the  distance  abeam  too  small, 
he  may  fail  to  clear  rocks  or  shoals  extending  seaward ;  and 
if  he  makes  it  too  large,  he  may  lengthen  his  voyage  unneces- 
sarily in  rounding  the  light. 

There  are  certain  pairs  of  angles  (S"SL  and  S"S'L)  which 
will  make  known  the  coming  distance  abeam  long  before 
the  ship  is  dangerously  near  the  light.  These  angles,  S"SL 
and  S"S'L,  are  called  "bearings  from  the  bow"  (see  p.  54), 
since  they  are  really  measured  from  the  ship's  bow  instead 
of  the  north.  If  the  two  bearings  from  the  bow  are  either 
of  the  following  pairs  : 

22°  and  34°,  32°  and  59°, 

27°  and  46°,  40?  and  79°, 

then  the  logged  distance  in  the  interval  between  the  two 
observations  is  the  distance  at  which  the  ship  will  pass  the 
light  abeam  if  she  continues  on  her  present  course.  This 
kind  of  observation  will  inform  the  navigator  whether  his 
course  is  safe  in  ample  time  to  change  it  if  necessary ;  and, 
since  in  this  case  no  bearings  are  marked  on  the  chart,  no 
attention  need  be  paid  to  compass  error. 

When  two  or  more  known  and  conspicuous  landmarks 
are  visible  from  the  ship,  it  is  possible  to  secure  a  fix  by 
means  of  "cross-bearings."  Observe  the  bearings  of  the 
objects  as  nearly  simultaneously  as  possible.  Allow  for 
compass  error  in  the  manner  just  explained.  Calculate 
for  each  object  a  reversed  bearing  by  adding  180°  to  its 
observed  bearing.  Draw  on  the  chart  through  each  object 


COASTWISE  NAVIGATION 


57 


a  pencil  line  having  the  proper  reversed  bearing  and  these 
lines  will  intersect  at  the  point  on  the  chart  where  the  ship 
is  located.  Figure  10 
illustrates  this  matter. 
L,  L',  L"  are  lights  or 
landmarks  ashore, 
visible  from  the  ship, 
and  also  printed  on 
the  chart.  The  ship 
is  at  S.  The  lines  in- 
tersecting at  S  repre- 
sent the  reversed 
bearings  of  L,  L',  L", 
as  observed  from  S. 
Only  two  lines  are  nec- 
essary; and  they 
should  be  chosen  so 
that  the  angle  be- 
tween them  is  as  near 


FIG.  10.  — Ship's  Position  by  Cross  Bearings. 


a  right  angle  as  possible,  if  high  accuracy  is  required  in  the 
fix.  The  third  object  and  line  merely  serve  as  an  additional 
check  or  safeguard  against  error. 

In  addition  to  the  foregoing  methods  of  locating  a  ship 
by  observations  of  objects  ashore,  there  is  a  way  to  avoid 
sunken  rocks  or  shoals  without  actually  locating  the  ship 
on  the  chart.  It  is  called  the  "  danger  angle,"  and  is  shown 
in  Fig.  11.  The  small  circle  is  supposed  drawn  on  the  chart 
around  a  rocky  shoal  K  which  must  be  cleared  by  the  ship 
traveling  along  the  course  SS'.  To  make  certain  of  clearing 
it  safely,  the  navigator  selects  two  visible  objects  ashore, 
and  shown  on  the  chart  at  L  and  L'.  He  draws  on  the 
chart  a  large  circle  passing  through  L  and  I/,  and  just  touch- 
ing the  dangerous  small  circle  at  T.  There  is  no  difficulty 
in  finding  the  center  of  the  large  circle,  because  it  must  be 
somewhere  on  the  line  PQ,  which  is  drawn  at  right  angles 
to  the  line  LL'  at  its  middle  point  P.  A  few  trials  with  a 


58 


NAVIGATION 


pair  of  compasses  will  locate  the  center.  Next,  the  two  lines 
LT  and  L'T  are  drawn.  Then  the  angle  LTL'  is  called  the 
danger  angle. 

Now  it  is  a  principle  of  geometry  that  if  we  select  other 
points  on  the  large  circle,  such  as  T'  and  T",  the  angles 


FIG.  11.  —  The  Danger  Angle. 

LT'L',  LT"L',  etc.,  will  all  be  equal,  and  will  contain  the 
same  number  of  degrees  as  the  danger  angle  LTL'.  It  fol- 
lows that  if  the  navigator  measures  from  the  deck  the  angle 
formed  by  two  lines  drawn  to  the  ship  from  L  and -I/,  and 
if  he  finds  it  equal  to  the  danger  angle  LTL',  as  measured 
on  the  chart  with  a  protractor  (p.  55),  he  then  knows  that 
the  ship  is  somewhere  on  the  large  circle,  and  is  therefore 
perhaps  too  near  the  small  dangerous  circle.  If,  on  the 
other  hand,  the  ship  is  entirely  outside  the  large  circle,  and 
therefore  surely  safe  from  the  dangers  of  the  small  circle, 


COASTWISE  NAVIGATION 


59 


N/l 


the  measured  angle  at  the  ship  between  the  objects  L  and 
U  will  always  be  smaller  than  the  danger  angle  LTL'. 

Angles  can  be  measured  from  the  deck  by  taking  compass 
bearings  of  L  and  L' .  The  difference  of  the  two  will  be  the 
deck  angle,  which  should  be  smaller  than  the  danger  angle 
measured  on  the  chart.  But  the  very  best  way  to  measure 
the  deck  angle  is  to  use  the  sextant,  an  angle-measuring 
instrument  to  be  described  later  (p.  61). 

The  danger  angle  can  also  be  used  when  it  is  necessary  to 
pass  between  a  sunken  danger  circle  and  the  shore.  The 
large  circle  is  then  drawn  through  L  and  L'  as  before,  but  in 
such  a  way  as  just  to  touch  the  inside  of  the  small  circle 
instead  of  the  outside.  To  pass  inshore  of  the  small  circle 
it  is  then  necessary  for  the  navigator 
to  keep  his  measured  deck  angle  larger 
than  the  danger  angle,  instead  of 
smaller. 

Navigators  also  use  at  times  a 
means  of  safety  known  as  the  "  danger 
bearing,"  illustrated  in  Fig.  12. 
There  is  but  one  charted  object  in 
sight  ashore  at  the  point  L.  The  ship 
at  S  must  steer  in  such  a  way  as  to 
avoid  sunken  rocks  at  K.  Evidently, 
she  must  pass  outside  the  line  SQ,  of 
which  the  bearing  from  the  north  is 
the  angle  NSQ,  which  can  be  meas- 
ured on  the  chart.  This  is  the  danger 
bearing,  and  the  ship's  course  SS',  to 

be  safe,  must  be  greater  than  the  danger  bearing.  In  the 
case  shown  in  the  figure,  the  danger  bearing  would  be  very 
useful  long  before  a  fix  could  be  had  by  means  of  bearings 
from  the  bow  or  bow-and-beam  bearings. 

Finally,  to  complete  this  part  of  our  subject,  it  is  neces- 
sary to  mention  "  soundings, "  which  are  a  method  of  feel- 
ing the  land,  even  when  it  cannot  be  seen.  By  means  of 


FIG.  12.  — The  Danger 
Bearing. 


60  NAVIGATION 

the  " lead-line"  the  mariner  can  ascertain  when  he  is  in 
shoal  water ;  and  as  depths  of  water  are  always  marked  on 
the  chart,  he  can  often  get  valuable  information  as  to  the 
ship's  position.  As  she  runs  along  her  course,  he  can  take 
a  "line  of  soundings"  and  upon  examining  the  chart  he 
will  often  find  but  a  single  possible  line  on  the  chart  where 
the  charted  depths  correspond  with  those  observed.  It 
follows  that  the  ship's  course  must  have  been  along  that 
line  on  the  chart ;  and  at  an  anxious  moment,  in  a  fog,  such 
a  check  will  be  a  great  relief  to  the  navigator.  Even  in 
the  ocean,  far  from  land,  it  is  possible  to  take  soundings 
with  the  "sounding  machine"  at  great  depths,  and  in  some 
parts  of  the  ocean  quite  accurate  locating  of  the  ship  will 
result.  Specimens  from  the  ocean  floor  can  also  be  brought 
up  by  attaching  some  sticky  grease  to  the  bottom  of  the 
lead,  and  at  times  these  specimens  also  give  information 
of  value,  for  the  charts  always  specify  the  kind  of  bottom 
existing  in  various  parts  of  the  ocean. 


.  CHAPTER   VI 
THE  SEXTANT 

WE  have  twice  made  reference  to  this  instrument  —  once 
(p.  5)  as  a  contrivance  for  ascertaining  by  observation  how 
high  the  sun  is  in  the  sky,  and  again  (p.  59)  in  the  measure- 
ment of  the  danger  angle.  These  two  uses  of  the  sextant 
are  not  inconsistent,  for  it  is  really  intended  for  the  measure- 
ment of  any  angle  (p.  8)  formed  at  the  observer's  eye  by 
two  lines  drawn  to  two  distant  objects.  In  the  case  of  the 
danger  angle  these  two  distant  objects  are  landmarks 
ashore;  in  the  case  of  the  sun  they  are  the  " horizon"  line 
(where  sea  and  sky  seem  to  meet),  and  the  sun  itself.  This 
height  of  the  sun  (or  of  any  star)  in  the  sky  is  called  its 
" altitude";  and  so  the  altitude  is  always  an  angle,  to  be 
measured  in  degrees  and  minutes.  The  point  directly  over- 
head is  the  "zenith";  the  angle  between  lines  drawn,  to 
horizon  and  zenith  is  90°,  or  a  right  angle.  An  altitude  of 
40°,  for  instance,  simply  means  that  the  distance  from  the 
horizon  to  the  sun  is  f$  of  the  total  distance  from  horizon 
to  zenith. 

Figure  13  will  give  an  idea  of  the  construction  of  the  sex- 
tant.1 The  essential  parts  are  two  small  silvered  mirrors, 
M  and  m;  a  telescope,  EK\  and  a  circle,  A  A,  engraved 
with  " graduations,"  by  means  of  which  angles  may  be 
measured  upon  it  in  degrees,  minutes,  and  seconds.  The 
mirror  m  and  the  telescope  EK  are  firmly  attached  to  the 
sextant ;  but  the  mirror  M  is  pivoted  in  such  a  way  that  it 

1  Quoted  in  part  from  Jacoby's  "Astronomy,  a  Popular  Hand- 
book," Macmillan,  1913;  reprinted  1915. 

61 


62 


NAVIGATION 


can  be  turned,  and  the  angle  through  which  it  is  turned 
measured  on  the  circle  by  means  of  the  index  CB.  When 
the  mirror  M  is  turned  until  it  is  parallel  to  the  fixed  mirror 
m,  the  circle  " reads"  or  indicates  0°,  because  the  angle  be- 
tween the  two  mirrors  is  then  0°.  In  all  other  positions 


FIG.  13.  — The  Sextant. 

of  the  mirror  M  the  circle  measures  the  angle  between  the 
two  mirrors.  P  and  Q  are  sets  of  colored  glasses,  which  can 
be  interposed  temporarily,  when  the  sun's  rays  are  so  bril- 
liant as  to  be  hurtful  to  the  observer's  eye.  R  is  a  small 
magnifying  glass,  pivoted  at  S,  intended  to  facilitate  the 
examination  of  the  index  CB.  At  C  and  B  are  shown  the 
"clamp,"  by  which  the  index  can  be  fastened  to  the  circle, 
and  the  " tangent  screw,"  or  " slow-motion  screw"  which 
will  adjust  it  delicately,  after  it  has  been  clamped.  /  and  F 
are  additional  telescopes  or  accessories. 

The  mirror  m  has  an  important  peculiarity.  The  silver- 
ing is  scraped  away  at  the  back  of  the  mirror  from  half  its 


THE   SEXTANT  63 

surface.  Thus  only  one  half  reflects ;  the  other  half  is 
simply  transparent  glass.  A  navigator  looking  into  the 
telescope  at  E  will  therefore  look  through  the  mirror  m  with 
half  his  telescope,  and  with  the  other  half  he  will  look  into 
the  mirror. 

Now  it  is  a  fact  that  half  a  telescope  acts  just  like  a  whole 
one.  If  a  person  using  an  ordinary  spy-glass  half  covers 
the  big  end  with  his  hand,  he  will  see  the  same  view  he  saw 
with  the  whole  glass.  Only,  as  half  the  "  light-gathering " 
power  is  cut  off,  this  view  will  be  fainter,  —  less  luminous. 
Applying  this  to  the  sextant  telescope,  it  is  clear  that  the 
observer  will  see  two  things  at  once :  with  half  the  telescope 
he  will  see  what  is  visible  through  the  mirror  m ;  and  with 
the  other  half  he  will  see  what  is  visible  by  reflection  from 
the  mirror  m. 

If  he  holds  the  sextant  in  such  a  position  that  the  telescope 
is  horizontal,  while  the  frame  of  the  instrument  is  vertical, 
he  will  see  the  visible  sea  horizon  with  half  the  telescope 
through  the  mirror  m.  If  the  other  mirror  M  is  then  turned 
to  the  proper  position,  it  is  possible  to  see  the  sun  in  the  sky 
at  the  same  time,  with  the  other  half  of  the  telescope,  the 
solar  rays  having  been  reflected  successively  from  both  mir- 
rors, M  and  m.  To  make  this  possible,  the  sextant  tele- 
scope must  be  aimed  at  that  point  of  the  sea  horizon  which 
is  directly  under  the  sun.  The  solar  rays  will  then  strike 
the  mirror  M  first ;  be  thence  reflected  to  the  silvered  part 
of  the  mirror  m;  and  finally  reflected  a  second  time  into 
the  telescope.  Therefore  the  observation  consists  in  so 
turning  the  movable  mirror  M,  that  the  sun  and  horizon 
can  be  seen  coincidently  in  the  telescope. 

The  angle  between  the  mirrors  can  then  be  measured 
on  the  circle ;  and  it  is  easy  to  prove  by  geometry  that  the 
angular  altitude  of  the  sun  will  be  twice  the  angle  between 
the  two  mirrors.  Thus  it  should  merely  be  necessary  to 
double  the  mirror  angle,  as  indicated  by  the  sextant  index, 
to  obtain  the  solar  altitude.  But  the  sextant  makers  always 


64  NAVIGATION 

save  the  navigator  the  trouble  of  doubling  the  angle  by  the 
simple  device  of  numbering  half  degrees  on  the  arc  A  A  as 
if  they  were  whole  degrees ;  so  the  angle  as  it  comes  from  the 
sextant  is  already  doubled  for  further  use.  The  mirror  m 
is  called  the  "  horizon  glass,"  because  the  navigator  looks 
through  it  at  the  horizon.  The  other  mirror  M  is  the  "index 
glass,"  because  it  is  attached  to  the  index  arm. 

When  the  sextant  is  used  for  non-astronomical  observa- 
tions, such  as  the  danger  angle,  the  frame  is  held  horizontally, 
instead  of  vertically,  as  in  observations  of  the  sun.  The 
telescope  is  aimed  at  the  left-hand  object  ashore,  and  that 
object  is  viewed  through  the  horizon  glass  m.  The  index 
glass  M  is  then  turned  until  light  from  the  right-hand  object 
is  also  brought  into  the  telescope,  after  successive  reflections 
from  the  two  mirrors  M  and  m.  The  two  objects  will  then 
be  seen  "  superposed,"  and  the  sextant  arc  will  give  the 
angle  between  two  lines  drawn  from  the  observer  on  board 
to  the  two  objects  ashore.  This  angle  should  be  smaller 
than  the  danger  angle  to  keep  the  ship  safely  off-shore  of 
sunken  dangers  (p.  59). 

Reading  the  sextant  circle,  or  ascertaining  from  it  the 
angle  that  has  been  measured,  is  accomplished  by  means  of 
a  "vernier."  This  is  a  short  circular  arc,  engraved  with 
graduations  resembling  those  on  the  sextant  circle,  attached 
to  the  index  CB  (fig.  13)  just  under  the  little  magnifier  R. 
It  is  so  placed  that  the  graduations  on  the  sextant  circle 
and  the  vernier  are  close  together  and  can  be  seen  at  the 
same  time  through  the  magnifier  R.  Figure  14  gives  an  idea 
of  the  vernier  and  a  part  of  the  sextant  circle  near  the  zero 
of  its  graduations.  Numbers  on  both  circle  and  vernier 
increase  toward  the  left.  On  the  circle,  the  largest  spaces, 
marked  by  long  lines,  are  whole  degree  spaces.  Each  is 
usually  divided  into  two  halves  of  30'  each  indicated  by 
shorter  lines,  and  these  are  again  subdivided  into  three 
small  spaces  of  10'  each.  The  divisions  on  the  vernier 
resemble  those  on  the  circle,  except  that  the  degree  spaces 


THE   SEXTANT 


65 


of  the  former  are  here  called  min- 
ute spaces,  and  the  10'  spaces  of 
the  former  are  called  10"  spaces. 
The  real  index  of  the  instru- 
ment is  the  zero  mark  on  the 
vernier,  sometimes  provided  with 
an  engraved  "arrow."  If  this 
falls  exactly  on  a  degree  mark  of 
the  circle,  say  the  1°  mark,  the 
reading  of  the  instrument  is  ex- 
actly 1°  0'  0".  If  it  falls  exactly 
on  a  small  line  of  the  circle,  say 
the  second  to  the  left  of  the  1° 
mark,  the  reading  is  exactly  1° 
20'  0".  But  if  it  falls  between  two 
of  the  small  lines,  say  between 
the  20'  and  30'  marks  to  the  left 
of  the  1°  mark  (as  shown  in  the 
figure),  the  reading  must  be  1° 
20'  and  a  "bit."  It  is  the  busi- 
ness of  the  vernier  to  estimate 
the  size  of  that  bit.  To  do  this 
look  along  the  vernier  until  you 
find  a  line  which  is  exactly  op- 
posite some  line  on  the  circle. 
There  will  always  be  such  a  line  : 
in  the  figure  it  is  the  6'  line  of  the 
vernier.  Pay  no  further  atten- 
tion to  noting  which  line  on  the 
circle  is  the  one  thus  "  exactly 
opposite";  it  matters  not  which 
line  it  is.  But  read  carefully  the 
number  on  the  vernier  belonging 
to  the  "exactly  opposite"  line 
you  have  found  there.  Being  on 
this  occasion  the  6'  line,  it  follows 


VA 


A/1 


66  NAVIGATION 

that  the  bit  is  6' ;  and  as  we  found  the  reading  to  be  1°  20' 
and  a  bit,  the  complete  reading  is  1°  20'  +  6'  =  1°  26'. 

If  the  vernier  line  that  happened  to  be  " exactly  opposite" 
was  not  one  of  the  ten  long  minute  lines,  but  fell  between 
two  of  them,  it  would  indicate  that  the  bit  was  made  up  of 
minutes  and  seconds,  instead  of  being  an  exact  number  of 
minutes.  For  each  space  the  " exactly  opposite"  vernier 
line  happens  to  lie  to  the  left  of  a  long  vernier  minute  line, 
10"  must  be  added  to  the  bit.  For  instance,  if  in  the  figure 
the  "exactly  opposite"  vernier  line  was  the  next  short  one 
to  the  left  of  the  6'  long  line,  the  bit  would  be  6''  10",  and  the 
complete  reading  1°  26'  10",  instead  of  1°  26'.  But  seconds 
are  not  really  required  when  observing  aboard  ship,  so  that 
it  will  be  sufficient;  in  using  the  vernier,  to  find  the  number 
of  the  long  vernier  line  that  comes  nearest  to  being  "  exactly 
opposite." 

It  will  also  be  noticed  in  the  figure  that  the  sextant  circle 
has  some  additional  graduations  to  the  right  of  the  0°  mark. 
These  are  called  "off  the  arc"  graduations,  and  it  is  some- 
times necessary  to  read  a  small  angle  upon  them,  measuring 
from  the  0°  mark  to  the  right  instead  of  the  left.  This  makes 
it  necessary  to  read  the  vernier  backwards,  calling  the  0' 
mark  of  the  vernier  10'  and  the  10'  mark  0'. 

This  backward  reading  of  the  vernier  offers  no  particular 
difficulty,  and  it  is  especially  useful  in  determining  by  ob- 
servation the  "index  error"  of  the  sextant.  We  have  seen 
(p.  62)  that  when  the  two  sextant  mirrors  are  parallel, 
the  index  should  read  0°  0'  0".  But  it  is  seldom  possible 
to  adjust  the  instrument  so  that  this  condition  will  be  satis- 
fied exactly ;  nor  would  the  adjustment  remain  perfect  very 
long.  A  better  plan  is  to  determine  by  observation  how 
much  the  reading  differs  from  0°  0'  0",  when  the  mirrors 
are  parallel.  This  difference  is  the  index  error,  and  must 
be  applied  as  a  correction  to  all  angles  observed  with  the 
instrument. 

It  is  easy  to  make  the  mirrors  parallel :   we  have  merely 


THE   SEXTANT  67 

to  sight  some  distant  well-defined  terrestrial  object  like  the 
gilt  ball  on  the  top  of  a  flagpole  (or  the  sea  horizon,  if  aboard 
ship  at  sea),  after  clamping  the  index  near  0°.  We  shall 
then  see  in  the  telescope  two  images  of  the  distant  object; 
one  by  direct  vision  through  the  unsilvered  part  of  the  hori- 
zon glass,  the  other  after  reflection  from  both  mirrors.  By 
means  of  the  tangent  screw,  the  observer,  with  his  eye  at 
the  telescope,  can  bring  these  two  images  together,  so  that 
they  will  appear  as  a  single  image.  Then  the  mirrors  will 
be  parallel,  and  the  vernier  should  read  0°  0'  0".  If  it  actually 
reads  0°  8',  for  instance,  instead  of  0°  0'  0",  it  means  that  the 
reading  is  8'  too  large  on  account  of  index  error ;  and  every 
angle  measured  with  that  sextant  at  that  time  will  be  8' 
too  large,  and  must  be  corrected  by  subtracting  8'  from  it. 

If,  on  the  other  hand,  the  reading  is  8'  "off  the  arc," 
when  it  should  be  0°  0',  the  instrument  reads  &'  too  small, 
and  any  angle  measured  with  it  must  be  corrected  by  adding 
8'  to  it. 

For  accurate  determination  of  the  index  error  (and  it 
should  be  checked  frequently),  navigators  prefer  to  observe 
the  sun,  or  at  night,  a  star.  If  a  star  is  used,  the  process 
is  the  same  as  just  described  for  a  flagpole  ball.  But  if 
the  sun  is  used,  a  slightly  different  method  is  required.  The 
sun,  as  seen  in  the  telescope,  shows  a  round  disk  of  con- 
siderable size,  and  it  is  not  possible  to 
superpose  the  two  images  accurately. 
Therefore  it  is  better  to  make  them 
just  touch,  as  shown  in  Fig.  15,  when 
they  are  said  to  be  "tangent"  to  each 
other.  This  must  be  done  successively 
in  two  positions,  AB  and  BA.  In 
other  words,  after  the  first  "tangency  "  FlG*  15-~Index  Error- 
has  been  observed,  the  tangent  screw  (B,  fig.  13)  is  manipu- 
lated until  the  image  A  passes  across  B  from  top  to  bottom, 
and  gives  a  new  tangency  in  the  second  position. 

Each  tangency  will  give  a  reading  of  the  vernier.     Unless 


68  NAVIGATION 

the  sextant  is  greatly  out  of  adjustment,  one  of  these  read- 
ings will  be  off  the  arc,  the  other  on  the  arc.  If  there  were 
no  index  error,  the  off-arc  and  on-arc  readings  would  be 
equal;  if  they  differ,  half  the  difference  is  the  index  error. 
If  the  off-arc  reading  is  the  larger,  all  altitudes  measured 
with  that  sextant  must  be  increased  by  the  amount  of  the 
index  error ;  and  if  the  on-arc  reading  is  the  larger,  all  such 
altitudes  must  be  similarly  diminished. 

The  following  is  an  example  of  an  index  error  determina- 
tion: 

On-arc  readings,  Off-arc  readings, 

31'  20"  33'  20" 

31  40  33  50 

30  50  34     0 

Means,    31'  17"  33'  43" 

The  difference  is  33'  43"  -  31'  17"  =  2'  26".  Half  the 
difference,  or  1'  13",  is  the  index  error ;  and  because  readings 
on  the  arc  are  the  smaller,  all  angles  read  with  this  instru- 
ment must  be  increased  by  1'  13",  or,  for  ordinary  purposes 
of  navigation,  by  1'. 

In  addition  to  certain  "  ad  justing  screws"  with  which 
the  index  error  can  be  reduced  when  it  becomes  unduly 
large,  means  are  provided  for  three  other  sextant  adjust- 
ments. These  are : 

1.  To  make  the  index  glass  perpendicular  to  the  frame  of 
the  instrument. 

2.  To  do  the  same  with  the  horizon  glass. 

3.  To  set  the  telescope  parallel  to  the  frame  of  the  instru- 
ment. 

These  adjustments  are  always  completed  by  the  maker 
before  a  sextant  is  sent  out,  nor  does  the  navigator  usually 
need  to  correct  them  himself.  But  it  is  important  to  know 
how  to  test  them  occasionally.  Perpendicularity  of  the 
index  glass  can  be  examined  by  looking  into  the  glass  very 
obliquely  with  the  index  set  near  0°.  It  is  then  possible  to 
see  the  inner  edge  of  the  sextant  circle  both  by  looking  at 


THE   SEXTANT  69 

it  directly,  past  the  edge  of  the  index  glass,  and  also  by  reflec- 
tion in  the  glass  itself.  The  inner  edge  of  the  circle  should 
form  a  continuous  line  when  so  examined,  if  the  glass  is 
perpendicular ;  but  if  it  is  inclined,  the  line  will  appear  broken, 
instead  of  continuous. 

Secondly,  perpendicularity  of  the  horizon  glass  can  be 
tested  at  the  same  time  the  index  error  is  determined  by 
observing  a  star  or  a  distant  terrestrial  point  (p.  67).  The 
index  glass  having  been  properly  adjusted  to  perpendic- 
ularity, the  two  mirrors  can  never  be  made  parallel  by 
moving  the  index,  unless  the  horizon  glass  is  also  properly 
perpendicular.  Any  existing  lack  of  adjustment  will  there- 
fore betray  itself  in  the  index  error  determination,  because 
the  two  images  of  the  star  or  distant  object  will  not  be  super- 
posed in  any  position  of  the  index. 

Thirdly,  the  parallelism  of  the  telescope  to  the  frame  of 
the  instrument  can  usually  be  best  tested  with  an  ordinary 
pair  of  "  calipers." 

Having  thus  described  the  sextant,  its  adjustments,  and  its 
use  from  the  deck,  we  have  still  to  explain  how  it  can  be  used 
ashore.    Sometimes  it  is  necessary  for  the 
navigator  to  make  observations  ashore, 
when  it  is  not  usually  possible  to  see  the 
horizon  line    (p.   61).      Recourse    must 
then  be  had  to  an  "artificial  horizon," 
which   is   simply   an   iron  basin  full  of 
mercury  covered  with  a  glass  roof.     The 
mercury  furnishes  an  almost  perfectly 
horizontal   mirror,   and   the   glass    roof 
prevents  wind  from  ruffling  the  mercury 
surface,  and  thus  destroying  the  mirror.     FlG.  IG.  —  Artificial' 
Figure  16  explains  the  principle  of  the  Horizon, 

artificial  horizon.  HH  is  the  mercury  mirror,  S  the  sun, 
and  X  the  sextant.  The  observer  aims  the  sextant  telescope 
at  the  mercury  where  he  can  see  a  reflection  of  the  sun.  He 
then  measures  with  the  instrument  the  angle  between  a  line 


70 


NAVIGATION 


drawn  to  the  sun  as  seen  reflected  in  the  mercury  and  another 
line  drawn  to  the  actual  sun  in  the  sky.  It  can  be  shown 
by  geometry  that  this  measured  angle  will  be  just  twice  the 
real  altitude  of  the  sun,  such  as  it  would  be  if  observed  from 
the  sea  horizon.  Therefore,  in  using  the  artificial  horizon, 
it  is  merely  necessary  to  divide  the  sextant  angle  by  2  to  ob- 
tain the  correct  altitude  of  the  sun. 

In  observations  of  this  kind  two  "suns"  are  seen  at  the 
same  time  in  the  telescope,  just  as  is  the  case  in  index  error 
observations  (p.  67) ;  whereas  in  observing  from  the  sea 
horizon,  the  telescope  shows  only  one  solar  image  and  the 
horizon  line.  When  there  are  thus  two  solar  images,  they 
must  be  brought  into  tangency,  just  as  we  have  already 
explained  for  index  error  (p.  67).  When  there  is  but  one, 
it  must  be  brought  into  tangency  with  the  visible  sea 
horizon  line. 

But  this  altitude  is  not  yet  ready  to  be  used  in  the  further 
calculations  for  obtaining  the  position  of  the  ship  in  latitude 

and  longitude.  Further  pre- 
paratory corrections  must  be 
applied,  in  addition  to  the 
index  error  (p.  66),  which  is 
always  the  first  correction  to 
receive  attention.  These  pre- 
paratory corrections  are : 

1.  "Dip"  of  the  sea  hori- 
zon, due  to  the  elevation  of 
the  navigator  on  the  ship's 
deck  above  the  surface  of  the 
sea.  Its  cause  is  shown  in 
Fig.  17.  C  is  the  center  of  the 

FIG.  17.— Dip  of  the  Horizon.         earth,  K  a  point  at  sea  level, 

and  0  the  navigator,  elevated 

a  distance  OK  above  the  sea.  OZ  is  the  direction  of  the  ze- 
nith (p.  61),  OS  the  direction  of  the  sun,  and  OH  a  horizontal 
line  from  0.  OT  is  a  line  drawn  through  0,  and  just  touch- 


THE   SEXTANT  71 

ing  the  sea  surface  at  Tf.  Evidently  OT  will  be  the  direc- 
tion of  the  sea  horizon,  where  sky  and  sea  seem  to  meet. 
Therefore,  the  altitude  of  the  sun,  as  measured  from  the 
visible  sea  horizon,  will  be  the  angle  SOT;  whereas  the  angle 
we  require  is  the  angle  SOH,  or  the  altitude  of  the  sun 
above  the  true  horizontal  line  OH.  Therefore  the  angle 
HOT  is  a  correction  for  dip  which  must  be  subtracted  from 
all  measured  altitudes,  and  the  amount  of  the  correction 
depends  on  the  height  of  the  navigator's  eye  above  the  sea 
surface. 

2.  " Refraction"  is  a  bending  of  the  light  rays  as  they 
come  down  to  us  from  the  sun  through  the  terrestrial  atmos- 
phere.    It  always  makes  the  sun  seem  higher  in  the  sky 
than  it  really  is,  giving  another  subtractive  correction  for 
the  observed  altitude.     The  bending  here  involved  is  due 
to  the  passage  of  the  sun's  light  rays  through  atmospheric 
strata  of  increasing  density  as  the   light   approaches  the 
earth's  surface. 

3.  "Parallax"  is  a  small  correction  which  must  be  added 
to  the  observed  altitude  of  the  sun.     In  strict  theory,  all  astro- 
nomic observations  are  supposed  to  be  made  from  the  earth's 
center  instead  of  its  surface  where  the  ship  floats ;   and  the 
small  parallax  correction  allows  for  this  minor  theoretic 
point.     In  the  case  of  star  observations  this  correction  is 
zero. 

4.  "Semidiameter"    is   a   correction    depending   on   the 
choice  by  the  navigator  of  a  particular  point  on  the  sun's 
disk  (p.  67)  for  observation.     The  sun's  altitude,  as  used 
in  the  further  calculations,  should  be  the  altitude  of  the  sun's 
center ;   but  it  is  impossible  to  locate  the  center  of  the  disk 
accurately  in  the  telescope,  so  the  navigator  always  observes 
the  lowest  point  of  the  disk.     This  is  called  the  "  lower 
limb"  of  the  sun. 

Beginners  sometimes  have  difficulty  in  distinguishing 
the  upper  from  the  lower  limb  in  the  telescope.  The  best 
way  to  do  this  is  to  focus  the  telescope  on  some  distant 


72  NAVIGATION 

object,  and  note  whether  it  appears  upside-down  in  the 
field  of  view.  If  so,  the  telescope  is  an  " inverting"  one, 
and  the  top  of  the  sun  must  be  observed,  as  it  appears  in 
the  telescope,  though  it  will  really  be  the  correct  (or  lower) 
limb,  because  of  inversion  by  the  telescope.  When  using  the 
artificial  horizon  with  an  inverting  telescope,  the  tangency 
must  be  made  by  bringing  the  bottom  of  the  mercury  image 
in  contact  with  the  top  of  the  other  image.  The  high-pow- 
ered telescopes  supplied  with  good  sextants  are  usually  in- 
verting telescopes. 

Evidently  the  measured  altitude,  as  it  comes  from  the 
sextant,  must  be  increased  by  the  amount  by  which  the  sun's 
center  is  higher  than  the  lower  limb,  and  this  is  the  sun's 
semidiameter.  The  index  correction,  together  with  the 
above  four  additional  corrections,  will  fully  prepare  a  meas- 
ured sextant  altitude  of  the  sun  for  further  use  in  naviga- 
tional calculations.  In  the  case  of  a  star,  which  appears 
in  the  telescope  as  a  point  of  light  only,  without  any  per- 
ceptible disk,  no  semidiameter  or  parallax  corrections  are 
required;  and  in  using  the  artificial  horizon  (p.  69),  no 
correction  for  dip  is  necessary,  either  for  the  sun  or  a  star. 

It  is  possible  to  arrange  these  various  corrections  in  con- 
venient tables.  Thus,  in  Table  6  (p.  247),  we  give  a  combi- 
nation of  corrections  2  (refraction),  3  (parallax),  and  4  (semi- 
diameter),  to  be  used  for  observations  of  the  sun's  lower 
limb,  and  the  same  combination  without  the  semidiameter 
and  parallax  1  to  be  used  for  star  observations.  It  will  be 
noticed  that  the  tabular  corrections  vary  for  different  values 
of  the  observed  altitude,  which  appears  in  the  left-hand  col- 
umn of  the  table.  This  variation  comes  mainly  from  the 
refraction  part  of  the  combined  correction,  for  the  refrac- 
tion is  much  greater  when  the  sun  or  star  is  observed  at  a 
low  altitude  near  the  horizon  than  it  is  at  a  high  altitude 
near  the  zenith.  At  the  foot  of  the  page  is  given  a  small 
supplementary  correction  depending  on  the  date  in  the  year. 
1  Which  leaves  refraction  only. 


THE   SEXTANT  73 

This  small  correction  is  not  important  in  navigation,  but  is 
given  here  for  the  sake  of  completeness.  It  arises  from  the 
semidiameter  part  of  the  combined  correction,  for  the  an- 
nual orbit  of  the  earth  around  the  sun  is  of  such  a  shape 
that  the  earth  is  nearer  the  sun  in  January  than  it  is  in  July, 
which  makes  the  sun  appear  bigger  in  January.  And  when  the 
sun  appears  big,  the  semidiameter  will  of  course  be  large  too. 

Table  7  gives  the  dip  of  the  sea  horizon,  the  number  in  the 
left-hand  column  being  the  height  (in  feet)  of  the  navigator's 
eye  above  sea  level.  This  will  be  the  height  of  the  ship's 
deck,  increased  by  the  height  of  the  man's  eye  above  the 
deck.  Unfortunately,  the  dip,  as  given  in  Table  7,  at  times 
varies  considerably  from  the  dip  as  it  actually  exists  at  the 
ship.  The  'cause  can  be  seen  from  Fig.  17  (p.  70),  where 
it  will  be  noticed  that  the  line  from  the  observer  at  0  to  the 
sea  horizon  at  T'  passes  very  near  the  surface  of  the  ocean. 
It  is  therefore  entirely  in  the  lowest  strata  of  the  terrestrial 
atmosphere,  and  there  quite  irregular  refractions  sometimes 
occur.  These  have  been  known  to  produce  errors  in  the  dip 
amounting  to  10'  or  20',  and  it  is  principally  the  existence 
of  these  unavoidable  errors  that  makes  it  unnecessary  to 
read  the  sextant  closer  than  the  nearest  minute  (p.  66), 
when  observing  from  the  deck.  But  when  observing  ashore 
with  the  artificial  horizon,  which  has  no  dip,  the  navigator 
may,  if  he  chooses,  read  seconds,  especially  if  he  intends  to 
use  in  his  further  calculations  the  "mean"  or  average  of 
a  considerable  number  of  observations. 

We  shall  now  give  an  example  of  the  complete  correction 
of  a  sextant  observation.  Suppose  the  angle  read  from 
the  sextant  was  30°  28',  the  index  error  (p.  68)  1',  addi- 
tive, height  of  observer's  eye  26  feet.  We  should  then 
have: 

observed  altitude,  lower  limb  =  30°  28' 

index  correction  =     +    1' 

correction  from  Table  6  (p.  247)  =     +  14' 

correction  from  Table  7  (p.  247)  =     -    5' 

corrected  altitude,  for  further  use  =  30°  38' 


74  NAVIGATION 

If  the  altitude  had  been  observed  ashore  with  an  arti- 
ficial horizon,  it  might  have  been  desirable  to  retain  seconds. 
The  calculation  might  then  have  been  as  follows : 

observed  double  altitude  (see  p.  70),  lower  limb  =  63°    0'  20" 

index  correction  (p.  68)  =       -f  1   13 

corrected  double  altitude  =63      1   33 

resulting  altitude  =31    30  46 

correction  from  Table  6  (interpolated)  =  +    14  31 

corrected  altitude,  for  further  use  =  31    45  17 


CHAPTER  VII 
THE  NAUTICAL  ALMANAC 

BEFORE  beginning  the  further  utilization  of  altitude  ob- 
servations in  our  navigation  calculations,  it  is  necessary  to 
understand  the  use  of  the  Nautical  Almanac.  This  is  an 
annual  publication,  issued  in  two  different  editions  by  the 
Nautical  Almanac  Office,  United  States  Naval  Observatory. 
Copies  can  be  obtained  from  the  Superintendent  of  Docu- 
ments, Washington,  D.  C.,  or  through  any  dealer  in  nautical 
supplies.  Navigators  do  not  need  the  larger  edition,  of  which 
the  title  is  " American  Ephemeris  and  Nautical  Almanac"; 
accordingly,  all  our  references  are  made  to  the  smaller  edi- 
tion for  the  year  1917.  Parts  of  certain  pages  from  that 
edition  are  reprinted  in  the  present  volume  for  convenience 
of  reference,  and  we  shall  give  a  somewhat  detailed  explana- 
tion of  the  almanac  page  29  (our  p.  76). 

Let  us  consider  the  date  Monday,  Dec.  17.  We  find  for 
that  date,  and  for  every  even  hour  (0*,  2*,  4*,  6h,  etc.)  of 
" Greenwich  Mean  Time"  (abbreviated  G.  M.  T.1),  two 
tabular  numbers  (p.  10)  called  " sun's  declination"  and 
"equation  of  time." 

To  understand  these  it  is  necessary  to  bear  in  mind  that 
the  kind  of  time  in  ordinary  use  is  "solar  time,"  as  kept  by 
the  sun.  The  "solar  day"  begins  at  "noon,"  called  Oft  in 
astronomic  navigation,  and  it  continues  through  twenty-four 
hours,  without  any  confusing  A.M.  and  P.M.  In  ordinary 
life  the  day  begins  twelve  hours  sooner,  at  midnight,  and 
runs  through  two  twelve-hour  periods  of  A.M.  and  P.M.  to 

1  The  reader  is  requested  to  note  carefully  this  abbreviation,  as 
it  will  be  used  very  frequently. 

75 


76 


NAVIGATION 


SUN,   DECEMBER,    1917.     From  Nautical  Almanac,  p.  29 


G.  M.  T. 

SUN'S  DEC- 
LINATION 

EQUATION 
OF  TIME 

SUN'S  DEC- 
LINATION 

EQUATION 
OF  TIME 

SUN'S  DEC- 
LINATION 

EQUATION 
OF  TIME 

Monday  17 

Tuesday  25 

Saturday  29 

h 

0                  / 

m     s 

0               / 

m       s 

0                  / 

m     s 

0 

-  23  21.3 

+  3  56.8 

-  23  24.7 

-0     1.6 

-  23  15.2 

-  1  59.7 

2 

23  21.5 

3  54.4 

23  24.6 

0     4.1 

23   14.9 

2     2.1 

4 

23  21.7 

3  51.9 

23  24.5 

0     6.5 

23   14.6 

2     4.6 

6 

23  21.9 

3  49.5 

23  24.4 

0     9.0 

23   14.3 

2     7.0 

8 

23  22.1 

3  47.0 

23  24.2 

0  11.5 

23   14.0 

2     9.4 

10 

23  22.2 

3  44.5 

23  24.1 

0  14.0 

23   13.7 

2  11.9 

12 

23  22.4 

3  42.1 

23  24.0 

0  16.5 

23   13.4 

2  14.3 

14 

23  22.6 

3  39.6 

23  23.8 

0  18.9 

23   13.1 

2  16.7 

16 

23  22.8 

3  37.1 

23  23.7 

0  21.4 

23   12.8 

2  19.1 

18 

23  22.9 

3  34.7 

23  23.5 

0  23.9 

23   12.5 

2  21.5 

20 

23  23.1 

3  32.2 

23  23.4 

0  26.4 

23   12.2 

2  24.0 

22 

23  23.2 

3  29.8 

23  23.2 

0  28.8 

23   11.9 

2  26.4 

H.  D. 

0.1 

1.2 

0.1 

1.2 

0.1 

1.2 

Tuesday  18 

Wednesday  26 

Sunday  30 

0 

-  23  23.4 

+  3  27.3 

-  23  23.1 

-  0  31.3 

-  23  11.6 

-  2  28.8 

2 

23  23.6 

3  24.8 

23  22.9 

0  33.8 

23  11.3 

2  31.2 

4 

23  23.7 

3  22.3 

23  22.7 

0  36.3 

23   11.0 

2  33.6 

6 

23  23.8 

3   19.9 

23  22.5 

0  38.7 

23   10.6 

2  36.0 

8 

23  24.0 

3  17.4 

23  22.4 

0  41.2 

23   10.3 

2  38.4 

10 

23  24.1 

3  14.9 

23  22.2 

0  43.7 

23  10.0 

2  40.9 

12 

23  24.3 

3  12.5 

23  22.0 

0  46.2 

23     9.7 

2  43.3 

14 

23  24.4 

3   10.0 

23  21.8 

0  48.6 

23     9.3 

2  45.7 

16 

23  24.5 

3     7.5 

23  21.7 

0  51.1 

23     9.0 

2  48.1 

18 

23  24.6 

3     5.0 

23  21.5 

0  53.6 

23     8.6 

2  50.5 

20 

23  24.8 

3     2.6 

23  21.3 

0  56.0 

23     8.3 

2  52.9 

22 

23  24.9 

3     0.1 

23  21.1 

0  58.5 

23     7.9 

2  55.3 

H.  D. 

0.1 

1.2 

0.1 

1.2 

0.2 

1.2 

Wednesday  19 

Thursday  27 

Monday  31 

0 

-  23  25.0 

+  2  57.6 

-  23  20.9 

-  1     0.9 

-  23     7.6 

-  2  57.7 

2 

23  25.1 

2  55.1 

23  20.7 

1     3.4 

23     7.2 

3     0.1 

4 

23  25.2 

2  52.6 

23  20.5 

1     5.9 

23     6.9 

3     2.4 

6 

23  25.3 

2  50.2 

23  20.3 

1     8.3 

23     6.5 

3     4.8 

8 

23  25.4 

2  47.7 

23  20.1 

1   10.8 

23     6.1 

3     7.2 

10 

23  25.5 

2  45.2 

23   19.8 

1   13.2 

23     5.8 

3     9.6 

12 

23  25.6 

2  42.7 

23   19.6 

1   15.7 

23     5.4 

3   12.0 

14 

23  25.7 

2  40.2 

23   19.4 

1   18.1 

23     5.0 

3  14.4 

16 

23  25.8 

2  37.8 

23  19.2 

1  20.6 

23     4.6 

3   16.7 

18 

23  25.9 

2  35.3 

23  19.0 

1  23.1 

23     4.3 

3   19.1 

20 

23  26.0 

2  32.8 

23  18.7 

1  25.5 

23     3.9 

3  21.5 

22 

23  26.1 

2  30.3 

23   18.5 

1  28.0 

-  23     3.5 

-  3  23.9 

H.  D. 

0.0 

1.2 

0.1 

1.2 

0.2 

1.2 

Thursday  20 

Friday  28 

o 

23  26.1 

+  2  27.8 

-  23  18.3 

-  1  30.4 

2 

23  26.2 

2  25.3 

23   18.0 

1  32.9 

4 

23  26.3 

2  22.8 

23  17.8 

1  35.3 

6 

23  26.3 

2  20.4 

23   17.5 

1  37.8 

8 

23  26.4 

2  17.9 

23   17.3 

1  40.2 

SEMIDIAMETER 

10 

23  26.5 

2   15.4 

23  17.0 

1  42.6 

12 

23  26.5 

2   12.9 

23   16.8 

1  45.1 

14 

23  26.6 

2  10.4 

23   16.5 

1  47.5 

Dec.    1 

16'26 

16 
18 
20 

23  26.6 
23  26.7 
23  26.7 

2     7.9 
2     5.4 
2     2.9 

23   16.3 
23   16.0 
23  15.7 

1  50.0 
1  52.4 
1   54.8 

11 
21 
31 

16'28 
16'29 
16'30 

22 

-  23  26.8 

+  2     0.4 

-  23   15.4 

-  1  57.3 

H.  D. 

0.0 

1.2 

0.1 

1.2 

NOTE.  —  The  Equation  of  Time  is  to  be  applied  to  the  G.  M.  T.  in  accordance  with 
the  sign  as  given. 


THE   NAUTICAL  ALMANAC  77 

the  following  midnight;  but  this  " civil  day/'  as  it  is  called, 
does  not  for  the  moment  concern  us. 

Solar  time,  as  kept  by  the  visible  sun,  is  a  very  incon- 
venient kind  of  time,  because  there  are  certain  peculiarities 
in  the  astronomic  motion  of  the  earth  which  make  these 
solar  days  of  unequal  length.  They  are  called  "  apparent 
solar  days"  and  the  corresponding  kind  of  time  is  " apparent 
solar  time." 

To  avoid  the  above  inconvenience,  an  imaginary  "mean 
sun"  and  a  "mean  solar  day"  have  been  invented.  The 
mean  sun  conforms  as  nearly  as  possible  to  the  average  per- 
formance of  the  visible  sun,  and  the  length  of  the  mean 
solar  day  is  the  average  of  all  the  apparent  solar  days  through- 
out the  year.  The  corresponding  kind  of  time,  kept  by  the 
mean  sun,  is  "mean  solar  time"  ;  and  this  is  the  kind  of  time 
recorded  by  all  our  watches  and  marine  chronometers  (p.  6). 

The  difference  between  these  two  kinds  of  solar  time  varies 
on  different  dates,  and  even  at  different  hours  on  the  same 
date.  It  is  this  difference  which  is  called  the  "equation  of 
time  "  and  which  is  one  of  the  tabular  numbers  in  the  nautical 
almanac  page  29  (our  p.  76). 

This  equation  of  time  is  of  great  importance  in  navigation, 
and  it  is  easy  to  see  how  page  29  of  the  almanac  may  be  used 
to  find  it.  Suppose,  for  instance,  we  wish  to  know  what  the 
equation  is  on  Dec.  17,  1917,  on  board  ship,  when  the  ship's 
chronometer  indicates  on  its  face  3  P.M.,  civil  time,  or  (which 
is  the  same  thing)  3h,  astronomical  time  (p.  75).  Ship's 
chronometers  are  always  set  to  Greenwich  mean  time,  so 
that  3*  by  the  chronometer  signifies  that  the  time  at  Green- 
wich was  3\ 

We  then  look  in  the  almanac  page  29  (our  p.  76),  and  find 
that  the  equation  was  +  3™  54*.4  at  2h,  G.  M.  T.,  and 
+  3m  5P.9  at  4A,  G.  M.  T.  Its  value  at  3h  must  be  half- 
way between  these  two,  or  -f  3m  53M5.  This  we  would 
call  -f  3m  53*.2,  so  as  to  avoid  the  use  of  hundredths  of 
seconds,  which  do  not  need  attention  in  navigation.  And 


78  NAVIGATION 

since  the  equation  is  merely  the  difference  between  the 
two  kinds  of  solar  time,  the  +  sign  means  that  it  must  be 
added  to  G.  M.  T.,  to  obtain  Greenwich  apparent  time,  in 
accordance  with  the  "Note"  at  the  foot  of  the  almanac 
page  29.  Consequently,  the  G.  M.  T.  by  chronometer  having 
been  3A  Om  0s,  the  Greenwich  apparent  time  at  the  same  in- 
stant was  3h  Om  0s  +  3™  53S.2  =  3h  3™  53a.2. 

It  will  be  noticed  that  the  process  we  have  here  used  for 
obtaining  the  equation  from  the  almanac  is  merely  an  inter- 
polation (see  p.  12).  Let  us,  as  another  example,  find  the 
equation  for  Sunday,  Dec.  30,  at  I0h  26W  A.M.,  civil  time  by 
chronometer,  and  we  have  purposely  here  retained  the 
civil  method  of  reckoning  time  to  make  certain  that  the 
reader  understands  the  difference  between  civil  and  astro- 
nomic (or  navigation)  time.  The  given  time  is  10*  26m  A.M., 
civil  time,  Dec.  30.  But  the  astronomic  Dec.  30  does  not 
begin  until  noon  (p.  75),  so  that  it  is  not  yet  Dec.  30  by 
astronomic  reckoning.  By  that  reckoning  it  is  really  only 
22^  26m  on  Dec.  29.  In  other  words,  when  the  civil  time  is 
P.M.,  as  in  the  first  example,  the  astronomic  time  is  the  same 
as  the  civil  time.  But  when  the  civil  time  is  A.M.,  as  in  the 
present  example,  the  astronomic  time  is  found  by  adding 
12*  to  the  civil  time,  and  deducting  1  from  the  date.  These 
complications  emphasize  the  advantage  of  the  astronomic 
count,  which  avoids  A.M.  and  P.M.  altogether. 

We  now  have  from  the  almanac  (p.  76) : 

equation  of  time,  Dec.  29,  22*,  G.  M.  T.  =  -  2m  26«.4, 

equation  of  time,  Dec.  30,     0*,  G.  M.  T.  =  -  2m  28».8  ; 

and  the  numbers  in  this  example  have  been  purposely  so 
chosen  that  the  above  two  tabular  values  of  the  equation 
(between  which  the  required  value  falls)  come  from  different 
dates  in  the  almanac.  This  creates  no  confusion,  for  these 
two  values  of  the  equation  are  really  consecutive  tabular 
numbers,  just  as  much  as  if  they  occurred  on  a  single  date. 
The  difference  between  the  two  values  of  the  equation  is 


THE   NAUTICAL  ALMANAC  79 

2a.4;  and  as  this  difference  corresponds  to  2h  in  the  left= 
hand  (or  argument)  column,  it  follows  that  the  difference 
for  lh  is  here  P.2.  This  is  the  change  of  the  equation  per 
hour  of  time;  it  is  called  the  " hourly  difference"  (abbre- 
viated H.  D.)  and  is  printed  in  the  almanac  at  the  foot  of 
each  daily  column. 

Now  we  want  the  equation  for  Dec.  29,  22*  26W,  by  the 
chronometer.  The  26m  must  next  be  changed  into  a  decimal 
fraction  of  an  hour.  26m  =  ff  of  an  hour  =  0\43.  So  the 
time  for  which  we  want  the  equation  becomes  Dec.  29, 
22A.43.  The  H.  D.  being  K2,  the  change  in  0^.43  will  be 
13.2  X  0.43  =  (K5.  The  almanac  shows  that  at  22h  the  equa- 
tion was  2m  26s. 4,  and  was  increasing  numerically.  There- 
fore, at  22\43,  it  was  2m  26S.4  +  05.5  =  2m  26S.9.  And  this 
number  has  the  minus  sign.  Therefore,  the  G.  M.  T.  being 
Dec.  29,  22h  26m,  the  Greenwich  apparent  time  at  the  same 
instant  will  be  Dec.  29,  22A  26™  -  2m  268.9  =  Dec.  29, 
22*  23-  33M. 

Most  of  these  minor  interpolation  calculations,  which  are 
here  set  forth  in  great  detail  for  the  benefit  of  the  beginner, 
can  be  made  with  sufficient  accuracy  by  a  skilled  navigator 
mentally. 

In  the  foregoing  two  examples  we  have  assumed  that  the 
chronometer  was  right,  but  these  instruments  practically 
never  run  quite  correctly.  Therefore,  before  leaving  port, 
navigators  always  have  their  chronometers  " rated"  by  a 
chronometer  expert;  and  when  the  instrument  is  returned 
to  the  ship  just  before  sailing,  a  "rate  card"  (or  "rate  paper") 
always  comes  with  it.  Let  us  suppose  that  in  the  present 
example  this  card  stated  that  the  chronometer  was  slow 
8m  228.5  l  on  Dec.  20,  at  noon,  and  was  "losing"  2  P.8  daily. 
The  8m  22". 5  would  then  be  the  "chronometer  error"  on 
Dec.  20 ;  and  the  1'.8  would  be  its  "daily  rate." 

1  This  number  is  here  purposely  chosen  much  larger  than  would 
ever  occur  in  practice. 

2  The  opposite  kind  of  "rate"  is  called  "gaining." 


80  NAVIGATION 

From  Dec.  20,  noon,  to  Dec.  30,  10h  26W  A.M.  is  an  interval 
of  9  days  22  hours  26  minutes.  This  interval  must  now  be 
reduced  to  a  decimal  of  a  day.  26m  =  f f  of  an  hour  =  0\43. 
The  interval  is  therefore  9*  22M3. 

But  22M3  =4£*  days  =  Od.93.  Therefore,  in  days,  the 
interval  is  9tf.93.  This  transformation  of  hours  and  minutes 
into  decimals  of  a  day  can  be  accomplished  with  less  trouble  by 
means  of  our  Table  8  (p.  248). 

Having  a  losing  rate  of  1s. 8  daily,  the  chronometer  lost 
1-.8  X  9.93  =  17S.9  in  the  interval  of  9.93  days.  And  as  it  was 
already  slow  8m  22S.5  on  Dec.  20,  it  was  slow  8m  22*.5  +17*.9 
=  8m  408.4  at  the  time  for  which  the  equation  is  required. 

Now  the  equation  was  required  for  Dec.  29,  22h  26m  by  the 
chronometer;  and  that  instrument  being  slow  8m  40S.4,  the 
correct  G.  M.  T.  was  :  Dec.  29,  22^  26  m  +  8m  408.4  =  Dec.  29, 
22h  34m  40S.4.  Turned  into  a  decimal  fraction  of  an  hour, 
this  becomes  Dec.  29,  22*.58,  instead  of  22\43,  as  we  found 
before,  when  the  chronometer  error  was  omitted  from  the 
calculation.  The  H.  D.  is  ls.2,  as  before,  and  the  change 
in  0^.58  =  P.2  X  0.58  =  0-.7.  Therefore,  at  22\5S  the 
equation  is  2m  265.4  +  Os.7  =  2m  27M.  This  still  has  the 
minus  sign,  so  that  the  correct  Greenwich  apparent  time 
becomes  Dec.  29,  22*  34™  40-.4  -  2m  27M  =  22*  32™  13'.3. 

All  the  above  calculations  have  been  carried  out  here  with 
unnecessary  accuracy.  There  would  be  no  harm  if  the  result 
were  in  error  by  a  few  tenths  of  a  second ;  and  it  is  this  cir- 
cumstance that  makes  it  possible  to  perform  these  inter- 
polations largely  mentally. 

In  the  foregoing  examples  no  account  was  taken  of  the 
ship's  location  on  the  ocean ;  yet  this  location  may  have  an 
indirect  influence  on  the  calculations.  To  understand  this, 
we  must  consider  for  a  moment  the  time-differences  which 
exist  between  different  places  on  the  earth.  The  sun  rises  in 
the  east  and  travels  across  the  sky  toward  the  west ;  so  that 
if  we  consider  two  places  like  Greenwich,  England,  and  New 
York,  for  instance,  the  sun,  because  of  this  motion  from  east 


THE   NAUTICAL  ALMANAC  81 

to  west,  will  pass  Greenwich  first.  Consequently,  when  it  is 
noon  in  New  York,  it  has  already  been  noon  in  Greenwich, 
and  is  afternoon  there.  Greenwich  time  is  therefore  always 
later  than  New  York  time.  The  same  is  true  of  any  other 
two  places ;  there  is  always  a  time-difference  between  them, 
and  the  easterly  place  has  the  later  or  " faster"  time. 

The  amount  of  such  time-difference  of  course  depends 
on  the  relative  location  of  the  two  places,  and  the  relation  is 
such  that  15°  of  longitude-difference  corresponds  exactly 
to  lh  of  time-difference.  Thus  Sandy  Hook,  which  is  in 
longitude  73°  50'  west  of  Greenwich,  has  a  time-difference 
from  Greenwich  of  4A  55 m  20s.  This  conversion  of  longitude 
into  time-difference  is  best  accomplished  by  means  of  our 
Table  9  (p.  249).  According  to  that  table : 

73°  =  4*  52"»    0* 

50'  3     20 

73°  50'  =  4*  55"  20* 

The  indirect  influence  of  such  time-differences  upon  the 
use  of  the  almanac  is  that  they  may  at  times,  especially 
when  they  are  large,  make  the  Greenwich  date  of  the  ob- 
servation different  from  the  date  on  board.  Thus  a  vessel 
off  Manila  Bay,  in  longitude  120°  east  of  Greenwich,  would 
have  her  local  time  8h  (120°)  later  than  Greenwich  time.  If 
a  sextant  observation  was  made  on  board  at  4  P.M.,  civil 
time,  on  a  Thursday,  the  chronometer  would  indicate  8A, 
and  it  would  be  8  A.M.  on  Thursday,  because  Greenwich  is 
Sh  earlier  than  the  ship.  This  8  A.M.  would  really  be  20h  of 
the  preceding  Wednesday  by  astronomic  time,  and  so  the 
almanac  date  used  would  be  one  day  earlier  than  the  date 
of  the  observation.  The  chronometer  will  always  give  the 
right  Greenwich  time,  but  the  navigator  must  be  very  care- 
ful to  interpolate  the  almanac  numbers  on  the  right  date. 

We  have  now  learned  how  to  ascertain  the  equation  of 
time  from  the  almanac,  and  how  to  use  it  for  transforming 
G.  M.  T.  into  Greenwich  apparent  time.  The  contrary 
transformation,  from  Greenwich  apparent  time  to  G.  M.  T., 


82  NAVIGATION 

can  be  made  by  applying  the  equation  in  the  opposite  way : 
subtracting  when  it  has  the  +  sign  in  the  almanac,  and  add- 
ing when  it  has  the  —  sign. 

The  great  importance  of  these  time  transformations  comes 
from  the  fact  that  sextant  observations  must  necessarily  be 
made  upon  the  visible  sun.  When  they  are  made  for  the 
purpose  of  calculating  the  local  time  on  board,  this  local 
time  will  therefore  necessarily  be  local  apparent  solar  time,  as 
kept  by  the  visible  sun.  At  the  instant  of  the  observation 
(p.  6),  the  chronometer  face  (corrected  for  error  and  rate) 
tells  us  the  G.  M.  T.  If  this  is  turned  into  Greenwich  ap- 
parent time  by  applying  the  equation,  we  have  only  to  com- 
pare the  Greenwich  and  the  ship's  apparent  times  to  get 
the  time-difference  between  the  ship  and  Greenwich.  This 
time-difference  can  then  be  turned  into  degrees  and  minutes, 
and  will  be  the  ship's  longitude.  Examples  of  this  calcu- 
lation will  be  given  in  detail  (p.  99).  It  is  also  worth 
noting  here  that  the  time-difference  between  any  two  places 
is  precisely  the  same,  quite  irrespective  of  the  kind  of  time 
in  which  it  is  counted. 

To  complete  our  explanation  of  the  almanac  page  29  (our 
p.  76),  it  remains  to  give  an  example  of  a  calculation  of  the 
sun's  declination.  This  is  an  angle  in  degrees  and  minutes, 
and  it  is  interpolated  just  like  the  equation  by  the  aid  of 
its  H.  D.  Thus,  for  Dec.  29,  22\58  (p.  80)  the  declination 
is  obtained  thus : 

Dec.  29,  22*,  declination          =  23°  11 '.9 

H.D.  (O'.l)  X  0A.58  0.1,  declination  decreasing ; 

by  subtraction,  at  22*.58,  dec.  =  23°  11 '.8, 

and  according  to  the  almanac,  this  declination  must  be  given 
the  minus  sign.  When  the  sign  should  be  +,  that  fact  is 
indicated  in  the  almanac.  The  use  of  the  declination  will 
be  explained  later;  the  accuracy  required  in  the  interpo- 
lation of  it  is  not  so  great  as  we  have  used  here,  for  the 
nearest  minute  suffices  in  practically  all  navigation  work. 
In  addition  to  the  sun's  declination,  navigators  require 


THE   NAUTICAL  ALMANAC 


83 


in  their  further  calculations  another  number  called  the  sun's 
"right  ascension"  (abbreviated,  R.  A.).  This  is  obtained 
from  pages  like  the  almanac  page  3  (reprinted  in  part  below). 
It  is  always  the  R.  A.  of  the  "mean  sun"  that  we  need, 
and  the  almanac  gives  it  for  Greenwich  mean  noon  of  each 
day  in  the  year.  When  needed  in  our  further  calcula- 
tions, it  is  of  course  always  required  for  the  exact  moment 
when  a  sextant  observation  was  made.  In  fact,  this  state- 
ment applies  also  to  the  equation  of  time  and  declination. 
They  must  always  be  interpolated  from  the  almanac  for  the 
moment  when  the  navigator  actually  observed  the  sun ;  and 

•    SUN,    1917.    From  Nautical  Almanac,  p.  3 


DAY 

OF 

MONTH 

RIGHT  ASCENSION  OF  THE  MEAN  SUN  AT  GREENWICH  MEAN  NOON 

July 

August 

September 

October 

November 

December 

h  m   s 

h  m    s 

h  m   s 

h  m   s 

h  m   s 

h  m   s 

1 

6  35  52.2 

8  38  5.5 

10  40  18.7 

12  38  35.3 

14  40  48.4 

16  39  5.1 

2 

6  39  48.8 

8  42  2.0 

10  44  15.2 

12  42  31.8 

14  44  45.0 

16  43  1.7 

3 

6  43  45.3 

8  45  58.6 

10  48  11.8 

12  46  28.4 

14  48  41.5 

16  46  58.2 

4 

6  47  41.9 

8  49  55.1 

10  52  8.3 

12  50  24.9 

14  52  38.1 

16  50  54.8 

5 

6  51  38.4 

8  53  51.7 

10  56  4.9 

12  54  21.5 

14  56  34.6 

16  54  51.3 

6 

6  55  35.0 

8  57  48.2 

11  0  1.4 

12  58  18.0 

15  0  31.2 

16  58  47.9 

7 

6  59  31.6 

9  1  44.8 

11  3  58.0 

13  2  14.6 

15  4  27.8 

17  2  44.5 

8 

7  3  28.1 

9  5  41.4 

11  7  54.5 

13  6  11.1 

15  8  24.3 

17  6  41.0 

9 

7  7  24.7 

9  9  37.9 

11  11  51.1 

13  10  7.7 

15  12  20.9 

17  10  37.6 

10 

7  11  21.2 

9  13  34.5 

11  15  47.6 

13  14  4.2 

15  16  17.4 

17  14  34.1 

11 

7  15  17.8 

9  17  31.0 

11  19  44.2 

13  18  0.8 

15  20  14.0 

17  18  30.7 

12 

7  19  14.3 

9  21  27.6 

11  23  40.8 

13  21  57.3 

15  24  10.5 

17  22  27.2 

13 

7  23  10.9 

9  25  24.1 

11  27  37.3 

13  25  53.9 

15  28  7.1 

17  26  23.8 

14 

7  27  7.4 

9  29  20.7 

11  31  33.9 

13  29  50.4 

15  32  3.6 

17  30  20.4 

15 

7  31  4.0 

9  33  17.2 

11  35  30.4 

13  33  47.0 

15  36  0.2 

17  34  16.9 

16 

7  35  0.6 

9  37  13.8 

11  39  27.0 

13  37  43.6 

15  39  56.8 

17  38  13.5 

17 

7  38  57.1 

9  41  10.4 

11  43  23.5 

13  41  40.1 

15  43  53.3 

17  42  10.0 

18 

7  42  53.7 

9  45  6.9 

11  47  20.1 

13  45  36.7 

15  47  49.9 

17  46  6.6 

19 

7  46  50.2 

9  49  3.5 

11  51  16.6 

13  49  33.2 

15  51  46.4 

17  50  3.2 

20 

7  50  46.8 

9  53  0.0 

11  55  13.2 

13  53  29.8 

15  55  43.0 

17  53  59.7 

21 

7  54  43.4 

9  56  56.6 

11  59  9.7 

13  57  26.3 

15  59  39.5 

17  57  56.3 

22 

7  58  39.9 

10  0  53.1 

12  3  6.3 

14  1  22.9 

16  3  36.1 

18  1  52.8 

23 

8  2  36.5 

10  4  49.7 

12  7  2.8 

14  5  19.4 

16  7  32.6 

18  5  49.4 

24 

8  6  33.0 

10  8  46.2 

12  10  59.4 

14  9  16.0 

16  11  29.2 

18  9  46.0 

25 

8  10  29.6 

10  12  42.8 

12  14  55.9 

14  13  12.5 

16  15  25.8 

18  13  42.5 

26 

8  14  26.1 

10  16  39.4 

12  18  52.5 

14  17  9.1 

16  19  22.3 

18  17  39.1 

27 

8  18  22.7 

10  20  35.9 

12  22  49.0 

14  21  5.6 

16  23  18.9 

18  21  35.6 

28 

8  22  19.2 

10  24  32.4 

12  26  45.6 

14  25  2.2 

16  27  15.4 

18  25  32.2 

29 

8  26  15.8 

10  28  29.0 

12  30  42.2 

14  28  58.8 

16  31  12.0 

18  29  28.7 

30 

8  30  12.4 

10  32  25.6 

12  34  38.7 

14  32  55.3 

16  35  8.6 

18  33  25.3 

31 

8  34  8.9 

10  36  22.1 

12  38  35.3 

14  36  51.9 

16  39  5.1 

18  37  21.9 

84 


NAVIGATION 


CORRECTION  TO  BE  ADDED  TO  R.  A.  M.  S.  AT  G.  M.   N.   FOR 

TIME   PAST   NOON 
From  Nautical  Almanac,  p.  3,  Continued 


TIME 

Qm 

6- 

12™ 

IS- 

24™ 

30™ 

36™ 

42™ 

48™ 

TIME 

h 

m   s 

m  s 

m   s 

m  s 

m   s 

m  s 

m  s 

m  s 

m  s 

h 

12 

1  58.3 

1  59.3 

2  0.2 

2  1.2 

2  2.2 

2  3.2 

2  4.2 

2  5.2 

2  6.2 

12 

13 

2  8.1 

2  9.1 

2  10.1 

2  1L1 

2  12.1 

2  13.1 

2  14.0 

2  15.0 

2  16.0 

13 

14 

2  18.0 

2  19.0 

2  20.0 

2  20.9 

2  21.9 

2  22.9 

2  23.9 

2  24.9 

2  25.9 

14 

15 

2  27.8 

2  28.8 

2  29.8 

2  30.8 

2  31.8 

2  32.8 

2  33.8 

2  34.7 

2  35.7 

15 

16 

2  37.7 

2  38.7 

2  39.7 

2  40.7 

2  41.6 

2  42.6 

2  43.6 

2  44.6 

2  45.6 

16 

17 

2  47.6 

2  48.5 

2  49.5 

2  50.5 

2  51.5 

2  52.5 

2  53.5 

2  54.5 

2  55.4 

17 

18 

2  57.4 

2  58.4 

2  59.4 

3  0.4 

3  1.4 

3  2.3 

3  3.3 

3  4.3 

3  5.3 

18 

19 

3  7.3 

3  8.3 

3  9.2 

3  10.2 

3  11.2 

3  12.2 

3  13.2 

3  14.2 

3  15.2 

19 

20 

3  17.1 

3  18.1 

3  19.1 

3  20.1 

3  21.1 

3  22.1 

3  23.0 

3  24.0 

3  25.0 

20 

21 

3  27.0 

3  28.0 

3  29.0 

3  29.9 

3  30.9 

3  31.9 

3  32.9 

3  33.9 

3  34.9 

21 

22 

3  36.8 

3  37.8 

3  38.8 

3  39.8 

3  40.8 

3  41.8 

3  42.8 

3  43.7 

3  44.7 

22 

23 

3  46.7 

3  47.7 

3  48.7 

3  49.7 

3  50.6 

3  51.6 

3  52.6 

3  53.6 

3  54.6 

23 

the  Greenwich  time  of  this  event  is  of  course  always  taken 
from  the  chronometer  (duly  corrected  for  error  and  rate). 

Thus,  if  the  R.  A.  of  the  mean  sun  is  required  for  Dec.  29, 
22*  34W  408.4,  G.  M.  T.  (p.  80),  we  find  from  the  almanac 
page  3  (our  p.  83)  that  the  R.  A.  of  the  mean  sun  at  Green- 
wich mean  noon  is  18*  29m  28'.?.1  This,  according  to  the  sup- 
plementary table  quoted  above  from  page  3,  must  be  increased 
by  a  correction  for  "time  past  noon."  In  this  case  the  time 
past  noon  is  22*  34W  40*.4.  The  tabular  correction  for  22*  30m 
is  3m  41«.8,  and  for  22*  36W  it  is  3m  42'.8.  Ours  falls  between 
these  two,  and  an  interpolation  makes  the  correction  3m  42S.6. 
Consequently,  the  R.  A.  of  the  mean  sun  for  Dec.  29,  22* 
34"  40-.4,  G.  M.  T.  is  18*  29m  28'.7  +  3W  42'.6  =  18*  33M  11-.3. 

It  will  be  noticed  that  the  small  supplementary  table 
(quoted  above  from  almanac  page  3)  only  runs  from  12*  to  24*. 
The  other  half  of  the  table,  from  0*  to  12*,  is  printed  on  the 
opposite  page  2  of  the  almanac.  There  is  also  another 
longer  table,  printed  near  the  end  of  the  almanac,  and  there 
called  Table.  Ill,  from  which  the  supplementary  correction 
can  be  taken  without  the  necessity  of  interpolation. 

It  is  not  absolutely  essential  that  the  navigator  learn  what 

1  Right  ascensions  are  always  thus  measured  in  hours,  minutes, 
and  seconds,  like  time,  and  they  are  counted  from  0*  to  24*. 


THE  NAUTICAL  ALMANAC  85 

the  words  "right  ascension"  and  "  decimation"  really  mean. 
But  for  the  benefit  of  those  who  are  curious  in  such  matters 
we  may  state  that  these  numbers  locate  the  position  of  the 
sun  (or  of  a  star)  on  the  sky.  The  sky  is  a  great  globe,  called 
by  astronomers  the  "celestial  sphere,"  and  all  heavenly 
bodies  are  located  upon  it  precisely  as  points  on  the  earth 
are  there  located  by  their  latitudes  and  longitudes  (p.  3). 
There  is  a  "celestial  equator"  with  two  "celestial  poles," 
corresponding  accurately  to  the  terrestrial  equator  and  poles. 
Declination  then  corresponds  exactly  to  latitude  on  the  earth, 
and  so  it  measures  the  distance  of  a  heavenly  body  from  the 
celestial  equator.  When  the  body  is  north  of  the  celestial 
equator,  the  declination  is  called  +. 

Right  ascension  similarly  corresponds  to  longitude ;  and  for 
the  beginning  point  of  right  ascensions  on  the  sky  there  is  a 
"celestial  Greenwich,"  which  is  called  the  "vernal  equinox." 

After  this  brief  digression  into  astronomy,  we  return  to 
our  subject.  We  have  seen  (p.  82)  that  observations  of 
the  sun  will  tell  us  only  apparent  solar  time,  because  it  is 
only  the  visible  sun  that  we  can  observe.  If  the  observations 
are  made  upon  a  star,  the  kind  of  time  is  different  from  any 
so  far  mentioned.  It  is  called  "sidereal  time,"  or  star  time. 

It  is  always  possible  to  change  mean  solar  time  into  sidereal 
time,  and  vice  versa,  by  a  simple  process  of  calculation ;  but 
the  only  change  of  this  kind  required  in  navigation  is  the 
transformation  of  G.  M.  T.  into  Greenwich  sidereal  time. 
To  make  this  transformation,  we  have  only  to  take  from  the 
almanac,  for  the  given  G.  M.  T.,  the  R.  A.  of  the  mean  sun, 
and  then  to  add  it  to  the  given  G.  M.  T. 

Thus,  to  find  the  Greenwich  sidereal  time  corresponding 
to  Dec.  29,  22*  34"  40-.4,  G.  M.  T.,  we  have  already  found 
(p.  84)  that  the  R.  A.  of  the  mean  sun  =  18*33m  11-.3 

To  this  must  be  added  the  given  G.  M.  T.     =  22  34   40.4 
Sum  =  corresponding  Greenwich  sidereal  time  =  17A17m51*.7 

1  The  number  of  hours  was  here  really  41* :  but  whenever  it  is 
larger  than  24*,  we  must  drop  or  reject  24*. 


CHAPTER  VIII 
OLDER  NAVIGATION  METHODS 

WE  shall  now  explain  in  detail  certain  standard  methods 
of  determining  a  ship's  latitude  and  longitude  by  means  of 
sextant  observations.  An  understanding  of  these  methods 
is  essential  to  a  proper  comprehension  of  the  newer  naviga- 
tional processes  to  be  described  later ;  and  the  older  methods 
are  in  fact  still  very  widely  used  at  sea,  although  most  re- 
cent authorities  believe  they  should  be  rejected  in  favor  of 
the  newer  procedure. 

The  simplest  of  these  older  processes,  and  the  one  most 
frequently  employed,  is  the  determination  of  the  ship's 
latitude  by  a  noon  or  " meridian"  observation  (" noon- 
sight")  of  the  sun's  altitude  (p.  61).  Now  the  sun  is 
higher  in  the  sky  at  noon  than  it  is  at  any  other  time  during 
the  day;  and  so  it  is  possible  to  get  the  noon-sight  by  be- 
ginning to  observe  the  sun  with  the  sextant  a  few  minutes 
before  noon,  and  continuing  the  observation  as  long  as  the 
sun's  altitude  is  increasing.  The  moment  it  begins  to 
diminish,  or  the  sun  to  "dip,"  as  sailors  say,  the  observation 
should  be  terminated,  and  the  vernier  read. 

The  altitude  thus  observed  will  be  an  altitude  of  the  lower 
limb  (p.  71) ;  and  before  it  is  used  further  it  must  be  fully 
corrected  for  index  error ;  for  refraction  parallax  and  semi- 
diameter  ;  and  for  dip ;  all  as  in  the  example  on  p.  73, 
where  the  observed  altitude  was  30°  28',  and  we  found  the 
corrected  altitude  to  be  30°  38'. 

Next,  the  sun's  declination  must  be  taken  from  the  al- 
manac, being  interpolated  for  the  Greenwich  time  of  the 

"86 


OLDER  NAVIGATION  METHODS  87 

observation,  as  in  the  example  on  p.  82,  where  we  found 
the  decimation  to  be  -  23°  12'  on  Dec.  29,  at  22*  34W  40'.4, 
G.  M.  T.  We  shall  suppose  the  above  altitude  30°  28'  to 
have  been  observed  at  the  Greenwich  time  stated,  so  as  to 
make  use  of  the  results  of  our  former  calculated  examples. 
Nor  is  there  any  inconsistency  in  supposing  a  noon  observa- 
tion to  have  been  made  at  22*  34W  40*.4.  For  the  noon 
observation  is  made  when  it  is  noon  on  board  ship,  while 
the  22*  34m  40-.4  is  the  G.  M.  T.  at  the  same  moment. 
The  difference  is  simply  the  time-difference  (p.  80)  between 
Greenwich  and  the  ship. 

The  calculation  of  the  ship's  latitude  is  now  made  by  the 
following  formula : 

Latitude  =  90°  +  Declination  -  Altitude. 

In  this  formula,  the  plus  sign  signifies  that  the  declination 
must  be  added;  and  the  minus  sign  signifies  that  the  altitude 
must  be  subtracted.  Furthermore,  it  is  most  important  to 
remember  that  if  the  declination  is  itself  a  "  minus  declina- 
tion," as  in  this  example,  the  addition  of  it  according  to  the 
formula  is  really  a  subtraction.  Or,  in  other  words,  and  in 
general,  whenever  a  formula  calls  for  an  addition,  and  the 
number  to  be  added  is  a  minus  number,  then  that  number 
must  be  subtracted  instead  of  added.  And  similarly,  if  the 
formula  calls  for  a  subtraction,  and  the  number  to  be  sub- 
tracted is  a  minus  number,  then  that  number  must  be  added 
instead  of  subtracted.  Two  minus  signs  neutralize  each  other. 

In  the  present  case  we  have,  omitting  seconds : 

90°  0' 

declination  =-23  12 

90°  +  declination  =     66  48 
altitude  =    30  38 

latitude  =    36  10 

In  considering  this  result  it  is  of  interest  to  inquire  where 
this  observation  really  locates  the  ship.  Now  we  have  not 
yet  stated  what  the  date  was,  on  board,  when  the  observa- 


88  NAVIGATION 

tion  was  made ;  but  we  have  given  the  G.  M.  T.  as  Dec.  29, 
22*  34OT  40*.4.  The  noon-sight  was  taken,  as  a  matter  of 
fact,  at  noon  on  Dec.  30,  or  at  the  moment  when  the  date 
Dec.  30  commenced  by  astronomic  reckoning.  Therefore 
the  ship's  time  was  later  than  the  Greenwich  time  by  about 
1*  25"*;  or  21°  15',  allowing  15°  to  1*  (p.  81) ;  and  the  ship 
was  (approximately)  in  21°  15'  east  longitude  from  Greenwich. 
This,  together  with  the  latitude  36°  10',  locates  the  ship  in 
the  Mediterranean,  south  of  Greece,  and  west  of  Candia. 

Although  we  have  thus  apparently  located  the  ship  com- 
pletely in  latitude  and  longitude  from  a  single  noon-sight, 
it  must  not  be  supposed  that  we  have  really  accomplished 
this.  The  noon-sight  is  only  suitable  for  ascertaining  the 
ship's  latitude ;  the  longitude  is  determined  so  inaccurately 
as  to  be  practically  useless.  The  reason  for  this  is  that 
near  noon  the  sun  changes  its  altitude  very  slowly,  because 
it  is  then  near  the  turning-point  where  its  upward  morning 
motion  is  about  to  become  a  downward  afternoon  motion. 
For  the  sun's  daily  motion  in  the  sky  is  upward  in  the  morn- 
ing and  downward  in  the  afternoon.  Near  noon  it  runs 
along  horizontally,  or  very  nearly  so,  for  several  minutes, 
so  that  its  altitude 'change  is  insignificant  during  that  time. 

It  follows  from  this  temporary  invariability  of  altitude 
that  we  cannot  determine  the  exact  moment  when  noon 
occurs  by  observing  altitude  changes  with  the  sextant.  But 
the  latitude  determination  is  not  affected;  because,  for 
the  latitude,  we  only  need  to  know  the  noon  altitude.  And 
if  we  happen  to  measure  it  a  little  too  soon  or  too  late,  on 
account  of  the  difficulty  of  fixing  the  moment  of  noon,  no 
harm  will  result,  because  the  altitude  very  near  noon  is  the 
same  as  it  is  at  noon  precisely,  as  we  have  just  seen. 

It  is,  in  general,  practically  impossible  to  determine  both 
latitude  and  longitude  from  a  single  observation.  To  deter- 
mine two  unknown  things,  at  least  two  different  observations 
must  be  made.  Nor  can  any  skillful  method  of  planning 
the  observation  overcome  this  fundamental  circumstance. 


OLDER  NAVIGATION  METHODS  89 

Returning  now  to  our  latitude  formula  (p.  87),  it  is 
necessary  to  modify  it  somewhat  in  case  we  happen  to  be  in 
the  tropics,  where  the  sun  may  pass  between  the  zenith  and 
the  celestial  pole.  Even  in  temperate  latitudes  a  celestial 
body  may  do  this,  if  we  happen  to  observe  a  star  instead  of 
the  sun.  In  such  a  case,  if  the  ship  is  in  the  northern 
hemisphere,  the  navigator  will  observe  the  sun's  altitude 
toward  the  north  at  noon  instead  of  toward  the  south,  as 
usual.  Furthermore,  in  very  high  northern  latitudes,  the 
" midnight  sun,"  as  it  is  called,  can  be  observed  toward  the 
north,  and  below  the  celestial  pole.  This  is  the  minimum 
altitude  during  the  day,  instead  of  the  maximum ;  but  it  is 
usable  for  a  latitude  determination.  Such  an  observation  is 
called  a  " lower  transit"  ;  and  it  can  often  be  observed  in  the 
case  of  stars  in  temperate  latitudes. 

If  we  now  remember  to  call  northerly  latitudes  and 
declinations  plus,  and  southerly  ones  minus,  we  have  the 
following  complete  set  of  formulas  for  the  present  problem, 
including  observations  in  both  hemispheres.  These  formulas 
are  so  arranged  that  we  can  easily  choose  the  right  formula, 
by  having  regard  to  the  -f-  and  —  signs.  But  the  right 
formula  once  chosen,  the  latitude  is  calculated  without 
marking  declinations  with  either  the  -f  or  —  sign. 

lat.1  and          if  lat.  greater  than  dec.,  lat.  =  90°  +  dec.  -  alt.  (1) 

dec.  both  +    if  dec.  greater  than  lat.,  lat.  =  dec.  +  alt.  -  90°  (2) 

or  both  -       if  lower  transit,  lat.  =  90°  +  alt.  -  dec.  (3) 

lat.  and  dec.,  1  lat  =  9QO  _  a]t  _  dec  (4) 
one  +,  one  -  j 

We  shall  now  give  some  more  examples ;  and  to  enable 
the  reader  to  follow  star  observations  correctly  we  reprint 
part  of  the  upper  halves  of  pages  94  and  95  (our  pp.  91,  92) 
of  the  Nautical  Almanac.  These  contain  the  right  ascensions 
and  declinations  (p.  85)  of  a  quantity  of  bright  stars  for 
various  dates  in  the  year.  These  numbers  are  correct  for  the 
moment  of  " upper  transit,"  which  is  the  moment  when  these 
1  Latitude  and  declination  are  abbreviated  lat.  and  dec. 


90  NAVIGATION 

stars  attain  their  maximum  altitudes.  This  event  cannot 
be  called  a  noon-sight  in  the  case  of  a  star ;  but  it  is  observable 
in  a  manner  perfectly  similar  to  a  solar  noon-sight. 

These  stellar  right  ascensions  and  declinations  change 
so  slowly  that  it  is  unnecessary  to  use  interpolation  when 
taking  them  from  the  almanac  pages. 

Proceeding  now  to  our  examples,  suppose  that  on  shore, 
at  Sandy  Hook  Light,  approximate  latitude  and  longitude 
40°  28'  N.,  74°  0'  W.,  on  Monday,  Dec.  17,  1917,  at  noon,  the 
double  altitude  of  the  sun's  lower  limb  was  observed  with  a 
sextant  and  artificial  horizon,  and  found  to  be  51°  48'.  The 
index  correction  required  by  the  sextant  was  +  4' ;  and  the 
G.  M.  T.  by  chronometer  was  4*  56m  at  the  moment  the 
observation  was  made.  Find  the  latitude.  We  have : 

Observed  double  altitude 51°  48'  (1) 

Index  correction +  4  (2) 

Adding  (1)  and  (2)  gives  corrected  double  altitude  51°  52'  (3) 

Halving  (3)  gives  observed  altitude 25  56  (4) 

Correction  from  Table  61  (p.  247) 4- 1£  (5) 

Adding  (4)  and  (5)  gives  fully  corrected  altitude 26°  10'  (6) 

Now  use  formula  (4)  (p.  89)  because  latitude  is  + 

and  declination  is  -  .  Write 90  0  (7) 

Subtracting  (6)  from  (7)  gives  90°  -  corrected  altitude . .  63  50  (8) 
Interpolate  declination  from  almanac  (p.  76).  This 

gives  declination 23  22  (9) 

Subtracting  (9)  from  (8)  gives  for  the  latitude 40  28  (10) 

With  regard  to  the  foregoing  example  it  is  worth  remark- 
ing that  if  there  had  been  no  available  chronometer  set  to 
Greenwich  time,  it  would  still  have  been  possible  to  calculate 
the  observation.  For  the  known  approximate  longitude, 
even  if  only  a  dead-reckoning  (p.  5)  longitude,  would  be 
quite  accurate  enough  to  make  possible  the  interpolation  of 
the  declination  from  the  almanac.  And  in  the  present 
example,  the  chronometer  was  only  used  in  getting  the 
declination  printed  in  line  (9)  above. 

1  Dip  correction  from  Table  7  not  needed  because  the  artificial 
horizon  was  used. 


OLDER  NAVIGATION  METHODS 


91 


APPARENT   PLACES   OF   STARS,    1917 

From  Nautical  Almanac,  p.  94 
FOR  THE  UPPER  TRANSIT  AT  GREENWICH 


No. 

CONSTELLA- 
TION NAME 

RIGHT  ASCENSION 

<N 

M 

(j 

Q 

c 

| 

•-» 

j>> 

"3 

•-» 

i 

< 

I 

I 

1 

1 

h   m 

s 

s 

s 

s 

a 

s 

a 

s 

s 

s 

1 

aAndrom. 

0    4 

6.3 

6.4 

7.4 

8.4 

9.4 

10.0 

10.3 

10.3 

10.0 

9.6 

2 

ft  Cassiop. 

0    4 

44.8 

44.4 

45.7 

47.3 

48.7 

49.7 

50.1 

49.9 

49.3 

48.4 

3 

PCeti 

039 

26.5 

26.3 

27.0 

28.0 

28.9 

29.7 

30.0 

30.1 

29.8 

29.5 

4 

5  Cassiop. 

1  20 

23.9 

22.3 

23.5 

25.1 

26.7 

28.1 

28.9 

29.2 

29.0 

28.2 

5 

«Urs.  Min. 

1  29 

89.0 

22.9 

45.5 

77.6 

112.8 

142.4 

161.2 

166.4 

155.3 

129.0 

6 

a  Eridani 

1  34 

39.1 

36.8 

37.6 

38.8 

40.3 

41.5 

42.3 

42.4 

41.9 

41.1 

7 

a  Arietis 

2    2 

31.0 

30.1 

30.8 

31.7 

32.7 

33.6 

34.3 

34.6 

34.7 

34.5 

8 

&  Eridani 

255 

8.8 

6.8 

7.2 

7.9 

9.0 

10.0 

10.8 

11.3 

11.4 

11.0 

9 

a  Persei 

3  18 

25.9 

23.9 

24.4 

25.5 

26.8 

28.2 

29.3 

30.2 

30.6 

30.5 

10 

«  Tauri 

431 

11.7 

10.3 

10.5 

11.0 

11.9 

12.8 

13.7 

14.5 

15.0 

15.2 

11 

ft  Orionis 

5  10 

35.1 

33.7 

33.7 

34.2 

34.7 

35.6 

36.5 

37.3 

37.8 

38.1 

12 

a  Aurigae 

5  10 

36.5 

34.5 

34.6 

35.2 

36.2 

37.5 

38.7 

39.9 

40.7 

41.1 

13 

y  Orionis 

5  20 

43.1 

41.7 

41.7 

42.1 

42.8 

43.7 

44.6 

45.4 

46.0 

46.4 

14 

e  Orionis 

5  32 

2.4 

1.0 

1.0 

1.3 

2.0 

2.8 

3.7 

4.5 

5.2 

5.5 

15 

a  Orionis 

550 

43.1 

41.8 

41.7 

42.0 

42.7 

43.5 

44.4 

45.3 

46.0 

46.4 

16 

«  Argus 

6  22 

9.2 

6.1 

5.5 

5.4 

6.0 

6.9 

8.1 

9.3 

10.2 

10.6 

17 

a  Can.  Maj. 

641 

31.6 

30.2 

30.0 

30.1 

30.6 

31.3 

32.2 

33.1 

33.8 

34.3 

18 

e  Can.  Maj. 

655 

24.1 

22.6 

22.2 

22.2 

22.6 

23.3 

24.2 

25.2 

26.0 

26.5 

19 

a  Can.  Min. 

734 

59.7 

59.0 

58.7 

58.8 

59.1 

59.8 

60.5 

61.5 

62.3 

63.0 

20 

/3  Gemin. 

740 

17.1 

16.3 

16.0 

16.0 

16.4 

17.1 

18.0 

19.0 

20.0 

20.8 

21 

«  Argus 

820 

51.4 

49.0 

48.0 

47.3 

47.2 

47.8 

48.9 

50.4 

51.8 

52.8 

22 

A  Argus 

9    4 

58.6 

57.9 

57.3 

56.9 

56.8 

57.1 

57.8 

58.9 

60.1 

61.0 

23 

ft  Argus 

9  12 

20.6 

18.1 

16.4 

15.1 

14.5 

14.8 

16.0 

17.9 

20.0 

21.7 

24 

a  Hydra 

9  23 

32.5 

32.6 

32.2 

32.0 

32.0 

32.3 

32.9 

33.7 

34.7 

35.6 

25 

a  Leonis 

10    3 

59.2 

59.7 

59.3 

59.1 

59.0 

59.2 

59.7 

60.5 

61.4 

62.4 

Had  it  been  thus  necessary  to  get  the  declination  without 
using  the  chronometer,  we  should  have  proceeded  as  follows  : 

Apparent  solar  time  of  noon  (p.  75) 0* 

Approximate  longitude  =  74°  0'  W.  =  (at  15°  to 

the  hour) 

Adding    (1)   and   (2)   (p.  81)  gives  approximate 

Greenwich  apparent  time 

Approx,  eq.  of  time,  Dec.  17,  at  4A  56m  (p.  76) 
Subtracting1    (4)    from    (3)    gives    approximate 

G.  M.  T 

Declination  interpolated  for  G.  M.  T.  in  line  (5)  is  —  23 

1  The  equation  is  additive  to  G.  M.  T.,  according  to  the  note  at 
the  foot  of  p.  76,  and  therefore  to  be  subtracted  from  Greenwich 
apparent  time. 


Oft     ow 

(1) 

4     56  W. 

(2) 

4     56 

+  4 

(3) 

(4) 

4     52 
23°   22' 

(5) 
(6) 

92 


NAVIGATION 


APPARENT   PLACES   OF   STARS,    1917 

From  Nautical  Almanac,  p.  95 
FOR  THE  UPPER  TRANSIT  AT  GREENWICH 


DECLINATION 

"Nn 

•\r  .  _    \ 

INO. 

I 

^5 

3 

1 

Ei 

1 

1 

1 

I 

w 

i 

SPECIAL  NAME 

MAG.* 

0 

/ 

/ 

/ 

/ 

/ 

i 

/ 

, 

/ 

1 

+  28 

38.2 

38.1 

38.0 

38.0 

38.0 

38.4 

38.5 

38.5 

38.5 

Alpheratz 

2.2 

2 

+  58 

41.9 

41.8 

41.7 

41.6 

41.5 

42.0 

42.1 

42.2 

42.2 

Caph 

2.4 

3 

-  18 

26.5 

26.5 

26.5 

26.4 

26.3 

26.0 

26.1 

26.2 

26.2 

Deneb  Kaitos 

2.2 

4 

+  59 

48.7 

48.7 

48.6 

48.4 

48.3 

48.6 

48.8 

48.9 

49.0 

Ruchbah 

2.8 

5 

+  88 

52.2 

52.2 

52.1 

52.0 

51.8 

52.0 

52.2 

52.4 

52.5 

Polaris 

2.1 

6 

-57 

39.7 

39.7 

39.6 

39.4 

39.2 

39.0 

39.2 

39.3 

39.4 

Achernar 

0.6 

7 

+  23 

4.5 

4.4 

4.4 

4.3 

4.3 

4.6 

4.7 

4.7 

4.7 

Hamal 

2.2 

8 

-40 

38.3 

38.3 

38.3 

38.2 

38.1 

37.7 

37.8 

38.0 

38.1 

Acamar 

3.0 

9 

+  49 

34.3 

34.3 

34.3 

34.2 

34.1 

34.3 

34.3 

34.4 

34.5 

1.9 

10 

+  16 

20.7 

20.7 

20.7 

20.7 

20.7 

20.8 

20.8 

20.8 

20.8 

Aldebaran 

1.1 

11 

-    8 

17.8 

17.8 

17.9 

17.9 

17.8 

17.5 

17.6 

17.7 

17.7 

Rigel 

0.3 

12 

+  45 

55.0 

55.1 

55.1 

55.1 

55.0 

54.9 

54.9 

55.0 

55.1 

Capella 

0.2 

13 

+    6 

16.6 

16.5 

16.5 

16.5 

16.5 

16.7 

16.7 

16.6 

16.6 

Bellatrix 

1.7 

14 

-    1 

15.2 

15.3 

15.3 

15.3 

15.3 

15.0 

15.1 

15.1 

15.2 

Alnitam 

1.8 

15 

+    7 

23.6 

23.5 

23.5 

23.5 

23.5 

23.7 

23.7 

23.6 

23.6 

Betelgeux 

1.0-1.4 

16 

-52 

39.0 

39.2 

39.3 

39.3 

39.2 

38.7 

38.7 

38.9 

39.1 

Canopus 

-0.9 

17 

-  16 

36.1 

36.2 

36.3 

36.3 

36.3 

35.9 

36.0 

36.1 

36.2 

Sirius 

-  1.6 

18 

-28 

51.5 

51.7 

51.7 

51.8 

51.7 

51.3 

51.4 

51.5 

51.6 

Adhara 

1.6 

19 

+    5 

26.3 

26.2 

26.2 

26.2 

26.2 

26.3 

26.2 

26.2 

26.1 

Procyon 

0.5 

20 

+  28 

13.6 

13.6 

13.6 

13.7 

13.7 

13.5 

13.5 

13.4 

13.4 

Pollux 

1.2 

21 

-59 

14.4 

14.6 

14.8 

14.9 

14.9 

14.4 

14.4 

14.5 

14.7 

1.7 

22 

-  43 

5.7 

5.9 

6.1 

6.2 

6.2 

5.8 

5.8 

5.9 

6.0 

2.2 

23 

-  69 

22.4 

22.6 

22.8 

22.9 

23.0 

22.5 

22.4 

22.5 

22.7 

Miaplacidus 

1.8 

24 

-    8 

17.9 

18.1 

18.1 

18.2 

18.2 

18.0 

18.0 

18.1 

18.2 

Alphard 

2.2 

25 

+  12 

22.2 

22.2 

22.2 

22.2 

22.2 

22.2 

22.1 

22.0 

21.9 

Regulus 

1.3 

1  When  the  number  in  this  column  is  very  small,  and  especially  when  it  is  minus, 
the  star  is  very  bright. 

It  is  further  to  be  noted  that  as  we  can  thus  obtain  the 
approximate  G.  M.  T.,  we  really  know  in  advance  the  approx- 
imate moment  when  the  observation  should  be  made.  So 
it  is  unnecessary  to  get  the  sextant  ready  a  long  time  before 
the  observation ;  and  it  is,  in  fact,  better  to  observe  at  the 
proper  predetermined  approximate  moment  rather  than  to 
wait  for  the  maximum  altitude  (p.  86). 

When  the  ship's  position  at  noon  can  be  predicted  with  fair 
approximation,  it  is  thus  possible  to  have  the  declination  and 
other  numbers  for  calculating  the  noon-sight  also  all  ready 


OLDER  NAVIGATION  METHODS  93 

in  advance,  so  that  the  latitude  will  be  immediately  available 
when  the  noon  altitude  has  been  read  from  the  sextant. 

We  shall  now  consider  the  following  example :  Off  St. 
Paul  de  Loando,  West  Africa,  approximate  latitude  8°  55' 
south,  approximate  longitude  12°  55'  east,  both  predicted 
in  advance  by  D.  R.  for  noon  on  Monday,  Dec.  31.  The 
altitude  of  the  sun's  lower  limb  is  to  be  measured.  Index 
correction  is  —  5'.  Height  of  eye,  26  ft. 
To  prepare  for  the  observation,  we  have,  as  before : 

Apparent  solar  time  of  noon 0*     O"1     (1) 

Approximate  D.  R.  longitude  =  12°  55'  east  =  (at  15°  to 

the  hour) 52  E.  (2) 

Subtracting  (2)  from  (1)  gives  approximate  Greenwich 

apparent  time,  Dec.  30 23       8       (3) 

Approximate   equation  of   time,   Dec.   30,  at  23*  8m 

(p.  76) -       3       (4) 

Subtracting  (4)  from  (3),  having  regard  to  —  sign  of 

(4),  gives  approximate  G.  M.  T 23     11       (5) 

The  navigator  will  then  make  the  observation  when  the 
G.  M.  T.  is  23*  llm,  as  indicated  by  the  chronometer,  duly 
corrected  for  error  and  rate.  This  would  of  course  also  be 
noon,  or  the  time  when  the  sun  attained  its  maximum  altitude 
for  the  day. 

Now  the  dials  of  chronometers  are  always  divided  into 
12  hours,  like  ordinary  watches,  although  navigators  count 
time  through  24  hours,  as  we  have  seen  (p.  75).  The 
reason  is  that  the  dial  would  be  overloaded  with  numbers 
if  there  were  24  hour  divisions.  Therefore,  when  we  speak 
of  the  chronometer  indicating  23*  llm,  it  must  be  under- 
stood that  the  actual  chronometer  indication,  or  "chro- 
nometer face,"  as  it  is  sometimes  called,  would  really  be 
11*  llm ;  only,  the  navigator  would  call  it  23*  llm,  astronomic 
time.  In  this  manner  civil  time  still  forces  its  way  into 
navigation,  by  way  of  the  chronometer  face. 

To  make  the  observation  at  the  prearranged  G.  M.  T.  by 
chronometer  it  is  not  desirable  to  carry  that  instrument  out 
into  the  sunlight,  where  the  observer  stands.  It  is  much 


94  NAVIGATION 

better  for  the  navigator  to  use  his  watch,  and  to  calculate  in 
advance  the  " watch  time"  of  the  observation.  To  do  this, 
it  is  merely  necessary  to  compare  the  watch  with  the  chro- 
nometer, and  thus  ascertain  how  much  the  watch  is  slow  or 
fast  of  the  chronometer.  This  amount  is  called  "chro- 
nometer minus  watch"  (abbreviated  C.  —  W.) ;  and  when  the 
watch  is  fast  of  the  chronometer,  C.  —  W.  is  marked  with  the 
minus  sign. 

To  obtain  the  watch  time  for  the  observation,  we  subtract 
C.  —  W.  from  the  G.  M.  T.  In  the  present  case  we  will 
suppose  the  watch  was  47 m  fast  of  the  chronometer.  Then 
C.  —  W.  =  —  47m.  To  get  the  watch  time  for  the  observa- 
tion we  must  subtract  —  47m  from  23h  llm.  Subtracting  a 
minus  number  is  equivalent  to  addition ;  and  so  the  watch 
time  is  23h  llm  +  47m  =  23h  58TO.  The  observation  would 
be  made  as  nearly  as  possible  2m  before  noon,  by  the  watch. 

In  this  connection  it  also  becomes  of  interest  to  inquire 
how  the  navigator's  watch  happened  to  be  47 TO  fast  of  the 
chronometer.  It  is  customary  aboard  ship  to  set  the  deck 
and  cabin  clocks,  and  all  watches,  to  the  ship's  local  apparent 
time  once  a  day  at  least.  To  do  this,  we  proceed  as  follows  : 

Take  from  chronometer  the  G.  M.  T.,  corrected  for  error  and  rate  (1) 
Apply  to  this  G.  M.  T.  the  eq.  of  time,  giving  Green'h  app.  time  (2) 
Apply  to  (2)  the  approximate  D.  R.  longitude,  adding  it  if  longi- 
tude is  E.,  which  gives  ship's  apparent  time (3) 

And  set  the  watch  to  the  time  (3). 

An  example  of  this  proceeding  can  be  had  from  the  data  on 
p.  93.  Suppose  the  watch  was  to  be  set;  and  the  chro- 
nometer time  was  23h  0TO.  We  should  then  prepare  to  set  the 
watch  in  about  5m,  when  the 

G.  M.  T.  by  chronometer  would  be 23*     5TO  (1) 

Chronometer  error  (corrected  for  rate)  say -  2  (2) 

Corrected  G.  M.  T.  by  chronometer,  (1)  +  (2) 23       3  (3) 

Equation  of  time  (p.  93) -  3  (4) 

Greenwich  apparent  time,  (3)  +  (4) 23      0  (5) 

Approximate  longitude  (p.  93) 52  E.  (6) 

Ship's  apparent  time,  (5)  +(6) 23     52  (7) 


OLDER  NAVIGATION  METHODS  95 

And  the  watch  would  be  set  to  23ft  52m,  when  the  chro- 
nometer face  was  23*  5m ;  or,  which  is  the  same  thing,  the 
watch  would  be  set  at  8m  to  12  when  the  chronometer  in- 
dicated 5  minutes  past  11. 

Sometimes  the  navigator  wishes  the  watch  to  be  correct 
by  ship's  apparent  time  at  noon,  but  desires  to  set  it  right 
half  an  hour  sooner,  so  as  to  be  free  at  noon  to  make  an 
observation.  In  that  case  he  calculates  by  D.  R.  what  the 
longitude  will  be  at  noon,  and  proceeds  practically  in  the 
same  way  as  before. 

Resuming  now  the  example  of  p.  93,  we  are  still 
off  St.  Paul  de  Loando,  and  at  2m  before  noon  by  the 
watch  (p.  94)  the  altitude  of  the  sun's  lower  limb  was 
measured. 

Suppose  it  was  found  to  be 75°    34'  (1) 

The  index  correction  was —  5  (2) 

Adding  (1)  and  (2),  with  regard  to  sign  of  (2),  gives 

corrected  altitude 75     29  (3) 

Correction  from  Table  6 +16  (4) 

Correction  from  Table  7,  for  26  ft.  height  of  eye —  5  (5) 

Adding  (3),  (4),  (5)  gives  corrected  altitude 75     40  (6) 

Formula  (2),  p.  89,  is  the  proper  one,  and  the  inter- 
polated declination,  disregarding  sign,  is 23       8  (7) 

Latitude,  by  formula,  is  (6)  +  (7)  -  90°,  or 8    48  (8) 

The  latitude  of  the  ship  is  therefore  8°  48'  south,  from  the 
above  noon-sight  observation.  The  difference  of  7'  from 
the  approximate  latitude  (p.  93)  might  easily  be  caused  by 
ocean  currents. 

Our  next  example  is  a  star  observation.  Position  of  ship 
by  D.  R.  March  23,  1917,  at  6a  30"*  ship's  time  is :  latitude 
40°  25'  N.,  longitude  46°  52'  W.,  so  that  she  is  near  the  turning 
point  in  the  southern  "lane  route"  followed  by  steamships 
bound  from  New  York  to  Fastnet  in  summer.  The  upper 
transit  (p.  89)  of  Sirius  was  observed;  and  the  sextant 
altitude  was  33°  7'.  Index  correction,  —  7' ;  height  of  eye, 
24ft. 


96  NAVIGATION 

The  calculation  is  as  follows  : 

Observed  altitude  of  Sirius 33°     7'  (1) 

Index  correction —  7  (2) 

Adding  (1)  and  (2),  having  regard  to  minus  sign  of  (2), 

gives  corrected  altitude 33       0  (3) 

Correction  Tables  6  and  7,  combined —  6  (4) 

Adding  (3)  and  (4)  gives  finally  corrected  altitude  ....  32     54  (5) 
Use  formula  (4),  p.  89,  because  latitude  is  +  and  decli- 
nation of  Sirius  — .     We  have 90°  (6) 

Subtract  (5)  from  (6),  giving  (90°  -  altitude) .  ... 57       6  (7) 

Declination  of  Sirius  (p.  92),  disregarding  sign,  is.  .  .   16     36  (8) 
Subtract  (8)  from  (7),  giving  (90°-  altitude -declina- 
tion), or  the  latitude 40     30  (9) 

Ship's  latitude  at  the  moment  of  observation  was  therefore 
40°  30'  N. 

In  making  such  a  star  observation,  it  is  of  course  possible 
to  follow  the  star  with  the  sextant  until  it  begins  to 
dip  (p.  86)  toward  the  horizon  exactly  as  we  have  ex- 
plained for  the  sun.  But  it  is  preferable  to  prepare  for  the 
observation  in  advance,  and  to  make  it  at  a  definite  prede- 
termined minute  by  the  navigator's  watch.  To  make  such 
preparation,  it  is  necessary  to  use  pages  96  and  97  of  the 
Nautical  Almanac,  parts  of  which  pages  are  reprinted  here 
(pp.  97,  98). 

The  almanac  page  96  gives  for  all  the  bright  stars  the 
G.  M.  T.  of  upper  transit  (p.  158)  at  Greenwich,  for  the  first 
day  of  each  month.  And  it  will  be  noticed  that  the  upper 
transit  is  here  called  "  meridian  transit,"  which  is  practically 
another  name  for  the  same  thing.  Almanac  page  97  (our 
p.  98)  then  gives  a  subtractive  correction,  applicable  to  the 
numbers  on  page  96,  to  make  them  correct  on  days  of  the 
month  other  than  the  1st. 

Another  small  correction  is  still  required  to  make  the 
numbers  right  in  the  approximate  D.  R.  longitude  of  the  ship, 
instead  of  the  longitude  of  Greenwich,  as  used  on  almanac 
page  96.  This  correction  is  subtractive,  if  the  ship  is  in  west 
longitude,  and  additive,  if  she  is  in  east  longitude ;  and  the 


OLDER  NAVIGATION  METHODS 


97 


MERIDIAN   TRANSIT   OF   STARS,    1917 

From  Nautical  Almanac,  p.  96 
GREENWICH  MEAN  TIME  OP  TRANSIT  AT  GREENWICH 


CONSTELLA- 
TION 
NAME 

MAG. 

z 
4 
H 

1 

« 
•< 

1 

(H 

•«! 

1 

1 

§ 

t 

Q 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

a  Androm. 

2.2 

5  21 

3  19 

1  29 

23  23 

21  25 

13  22 

11  24 

9  22 

7  24 

/3  Cassiop. 

2.4 

5  22 

3  20 

1  30 

23  24 

21  26 

13  22 

11  24 

9  22 

7  24 

PCeti 

2.2 

5  56 

3  54 

2  4 

f  0   2) 

)  23  68  S 

22  0 

13  57 

11  59 

9  57 

7  59 

8  Cassiop. 

2.8 

6  37 

4  35 

2  45 

0  43 

22  41 

14  38 

12  40 

10  38 

8  40 

a  Urs.  Min. 

2.1 

6  47 

4  45 

2  54 

0  52 

22  50 

14  49 

12  51 

10  49 

8  51 

a  Eridani 

0.6 

6  51 

4  49 

2  59 

0  57 

22  55 

14  52 

12  54 

10  52 

8  54 

a  Arietis 

2.2 

7  19 

5  17 

3  27 

1  25 

23  23 

15  20 

13  22 

11  20 

9  22 

0  Eridani 

3.0 

8  12 

6  10 

4  20 

2  18 

0  20 

16  12 

14  14 

12  12 

10  14 

a  Persei 

1.9 

8  35 

6  33 

4  43 

2  41 

0  43 

16  35 

14  38 

12  36 

10  38 

a  Tauri 

1.1 

9  47 

7  46 

5  55 

3  54 

1  56 

17  48 

15  50 

13  48 

11  50 

/3  Orionis 

0.3 

10  27 

8  25 

6  35 

4  33 

2  35 

18  27 

16  29 

14  28 

12  30 

a  Aurigse 

0.2 

10  27 

8  25 

6  35 

4  33 

2  35 

18  27 

16  29 

14  28 

12  30 

y  Orionis 

1.7 

10  37 

8  35 

6  45 

4  43 

2  45 

18  37 

16  39 

14  38 

12  40 

«  Orionis 

1.8 

10  48 

8  46 

6  56 

4  54 

2  56 

18  49 

16  51 

14  49 

12  51 

a  Orionis 

1.0-1.4 

11  7 

9  5 

7  15 

5  13 

3  15 

19  7 

17  9 

15  7 

13  9 

a  Argus 

-0.9 

11  38 

9  36 

7  46 

5  44 

3  46 

19  39 

17  41 

15  39 

13  41 

a  Can.  Maj. 

-1.6 

11  57 

9  55 

8  5 

6  3 

4  5 

19  58 

18  0 

15  58 

14  0 

eCan.  Maj. 

1.6 

12  11 

10  9 

8  19 

6  17 

4  19 

20  12 

18  14 

16  12 

14  14 

a  Can.  Min. 

0.5 

12  51 

10  49 

8  59 

6  57 

4  59 

20  51 

18  53 

16  52 

14  54 

/3  Gemin. 

1.2 

12  56 

10  54 

9  4 

7  2 

5  4 

20  57 

18  59 

16  57 

14  59 

«  Argus 

1.7 

13  36 

11  34 

9  44 

7  42 

5  44 

21  37 

19  39 

17  37 

15  39 

A  Argus 

2.2 

14  20 

12  19 

10  28 

8  27 

6  28 

22  21 

20  23 

18  21 

16  23 

0  Argus 

1.8 

14  28 

12  26 

10  36 

8  34 

6  36 

22  28 

20  30 

18  28 

16  31 

a  Hydrse 

2.2 

14  39 

12  37 

10  47 

8  45 

6  47 

22  40 

20  42 

18  40 

16  42 

a  Leonis 

1.3 

15  19 

13  17 

11  27 

9  25 

7  27 

23  20 

21  22 

19  20 

17  22 

amount  of  it  is  10*  for  every  15°  in  the  ship's  longitude. 
After  it  has  been  applied,  the  result  will  be  the  ship's  mean 
solar  time  of  the  star's  upper  transit. 

As  an  example,  let  us  take  the  preparation  for  the  fore- 
going observation  of  Sirius,  or  a  Can.  Maj.     We  have : 
G.  M.  T.  of  upper  transit,  March  1,  from  almanac 

page  96  above 8*     5m  (1) 

Correction    for  23d    day  of    month,   from   almanac 

page  97  (our  p.  98) -  1     27     (2) 

Correcting  (1)  with  (2),  having  regard  to  —  sign  of  (2)  6  38  (3) 
Further  correction  for  longitude  46°  52'  W.,  at  10*  per 

15°  of  longitude,  approximately 1     (4) 

Subtracting  (4)  from  (3)  gives  ship's  mean  solar  time 

of  the  observation 6     37     (5) 


98 


NAVIGATION 


MERIDIAN    TRANSIT    OF    STARS,    1917 
From  Nautical  Almanac,  p.  97 

CORRECTIONS  TO  BE  APPLIED  TO  THE  MEAN  TIME  OF  TRANSIT  ON 
THE  FIRST  DAY  OF  THE  MONTH,  TO  FIND  THE  MEAN  TIME  OF 
TRANSIT  ON  ANY  OTHER  DAY  OF  THE  MONTH 


DAY  OP 
MONTH 

CORRECTION 

DAY  OP 

MONTH 

CORRECTION 

DAY  OF 
MONTH 

CORRECTION 

h        m 

h       m 

h       m 

1 

-0       0 

11 

-0     39 

21 

-  1     19 

2 

0       4 

12 

0     43 

22 

1     23 

3 

0       8 

13 

0     47 

23 

27 

4 

0     12 

14 

0     51 

24 

30 

5 

0     16 

15 

0     55 

25 

34 

6 

-0     20 

16 

-0     59 

26 

-        38 

7 

0     24 

17 

1       3 

27 

42 

8 

0     28 

18 

1       7 

28 

46 

9 

0     31 

19 

1     11 

29 

50 

10 

0     35 

20 

1     15 

30 

54 

11 

-0     39 

21 

-  1     19 

31 

-  1     58 

NOTE.  If  the  quantity  taken  from  this  Table  is  greater  than  the 
mean  time  of  transit  on  the  first  of  the  month,  increase  that  time 
by  23h  56W  and  then  apply  the  correction  taken  from  this  Table. 


The  actual  observation  was  made  at  6A  30m,  ship's  time, 
as  indicated  by  the  navigator's  watch.  The  difference  of 
7m  between  6*  30W,  and  6*  37W  in  line  (5)  above,  is  due  to  the 
equation  of  time  (p.  77),  which  is  7m  on  March  23.  This 
7W,  if  applied  (with  its  proper  sign  from  the  almanac)  to 
line  (5)  above,  will  give  the  ship's  apparent  time;  and  we 
have  seen  that  watches  and  clocks  on  board  are  usually 
kept  set  to  apparent  and  not  mean  ship's  time  (p.  94). 

To  complete  this  part  of  our  subject,  we  have  still  to  con- 
sider a  few  additional  points  of  interest.  For  instance,  a 
star  chosen  for  observation  may  be  one  of  the  planets : 
Mars,  Jupiter,  or  Saturn.  These  look  like  very  bright  stars 
in  the  sextant  telescope;  and  calculations  depending  on 
them  are  similar  to  those  described  for  stars.  The  planetary 
declinations  and  the  G  M.  T.'s  of  their  upper  transits  are 
given  in  the  .almanac,  but  not  on  the  pages  reprinted  here. 


OLDER  NAVIGATION  METHODS  99 

The  moon  is  now  so  rarely  observed  that  we  have  not  given 
examples  of  lunar  observations. 

Sometimes  an  " ex-meridian"  observation  of  the  sun  or 
a  star  is  made  at  a  time  very  near  the  upper  transit,  on  a 
day  when  the  actual  transit  observation  could  not  be  secured 
because  of  clouds.  There  are  special  tables l  for  calculating 
observations  of  this  kind ;  but  we  have  not  included  them 
here  because  all  such  observations  can  be  satisfactorily 
treated  by  a  new  general  method  to  be  explained  later 
(p.  108). 

Having  now  fully  treated  the  older  standard  method  of 
determining  the  ship's  latitude,  let  us  next  consider  the  older 
way  of  obtaining  the  longitude.  This  cannot  be  done  when 
the  sun  (or  a  star)  is  near  its  maximum  altitude,  as  already 
explained  (p.  88).  The  most  favorable  opportunity  occurs 
when  the  observed  object  bears  (p.  44)  east  or  west ;  but 
it  is  not  always  possible  to  get  the  observation  on  such  a 
bearing.  In  that  case,  the  longitude  observation,  often 
called  a  " time-sight,"  must  be  taken  when  the  sun  is  near 
the  desired  bearing,  but  always  avoiding,  if  possible,  observa- 
tions at  very  low  altitudes.  And  if  a  very  low  altitude  has 
been  observed  in  an  emergency,  it  can  sometimes  be  checked 
by  a  later  observation  at  a  better  altitude. 

The  principle  on  which  the  time-sight  depends  is  simple. 
Calculations  based  on  the  measured  altitude  make  known 
the  ship's  mean  time  at  the  moment  of  observation.  At 
the  same  moment  the  chronometer  face  (p.  93),  duly  cor- 
rected for  error  and  rate,  tells  us  the  G.  M.  T.  The 
difference  between  the  two  times  then  grves  us  the  longitude 
(see  p.  82). 

The  calculations  for  this  problem  are  made  by  means  of 
Table  4  (trigonometric  logarithms)  and  Table  10  ("  haver- 
sines").  These  haversines  (abbreviated  hav.)  are  really 
additional  trigonometric  logarithms;  and  Table  10  gives 
in  every  case  not  only  the  haversine  itself,  which  is  really 
1  Tables  26  and  27  of  Bowditch's  "  Navigator,"  for.  instance. 


100  NAVIGATION 

a  logarithm,  but  also,  in  the  adjoining  heavy  type  col- 
umns, the  number  (abbreviated  No.)  of  which  the  haver- 
sine  is  the  log.  This  additional  heavy  type  number  is  not 
given  throughout  the  entire  table,  but  only  when  necessary 
for  working  Sumner  line  calculations  (see  Chapter  IX, 
p.  108).  It  is  not  needed  in  working  time-sights. 

The  argument  (p.  10)  of  the  haversine  table  is  a  double 
argument,  not  to  be  confounded  with  the  pairs  of  arguments 
already  explained  (p.  11).  In  the  haversine  table,  the  argu- 
ment is  generally  given  in  degrees  and  minutes,  as  well  as 
(for  convenience)  in  hours  and  minutes  of  time,  allowing 
the  usual  15°  to  each  hour,  etc. 

We  shall  now  solve  our  time-sight  problem  for  the  sun; 
and  in  doing  so  shall  make  use  of  two  angles  not  hitherto 
employed:  the  " polar  distance"  (abbreviated  p),  and  the 
"half  sum"  (abbreviated  s).  We  shall  also,  for  brevity, 
indicate  the  ship's  apparent  solar  time  by  T.  Then  we 
have  the  following  formulas : 

If  lat.  and  dec.  are  both  +  or  both  —  .  .  p  =  90°  —  dec.  (1) 

If  lat.  and  dec.  are  one  +  and  one  —  .  . .  p  =  90°  +  dec.  (2) 

In  every  case s  =  f  (alt.  +  lat.  +  p}  (3) 

If  time-sight  was  made  before  noon,  ship's  time, 

hav.  (24*-  T)  =  sec  lat.  +  esc  p  +  cos  s  +  sin  (s  -  alt.)  (4) 
If  time-sight  was  made  after  noon,  ship's  time, 

hav.  T  =  sec  lat.  +  esc  p  +  cos  s  +  sin  (s  —  alt.)  (5) 

In  using  these  formulas,  we  have  to  choose  between  (1) 
and  (2),  and  also  between  (4)  and  (5).  Formula  (3)  is 
always  used.  No  attention  need  be  given  to  the  signs 
of  the  declination  *or  latitude  except  in  choosing  between 
formulas  (1)  and  (2)  for  calculating  p;  and  in  choosing 
between  (4)  and  (5),  we  have  merely  to  note  whether  the 
time-sight  was  taken  in  the  forenoon  or  afternoon  by  ship's 
time. 

We  also  desire  to  emphasize  especially  that  these  formulas 
presuppose  the  latitude  to  be  known.  This  is  merely 
another  application  of  the  principle  (p.  88)  that  "both  lati- 


OLDER  NAVIGATION  METHODS  101 

tude  and  longitude  cannot  be  determined  from  a  single 
observation.  It  follows  that  in  using  this  method  we  must 
first  determine  the  latitude  by  a  noon-sight  before  we  can 
calculate  the  time-sight  for  longitude.  If  the  time-sight 
was  taken  in  the  afternoon,  the  noon-sight  will  naturally 
have  preceded  it,  and  the  ship's  latitude  at  noon  will  be 
known.  This  noon  latitude  must  then  be  carried  forward 
to  the  moment  of  the  afternoon  time-sight  by  D.  R.  methods 
(p.  7) ;  and  the  latitude  thus  obtained  must  be  used  for 
calculating  the  time-sight. 

But  if  the  time-sight  was  a  forenoon  observation,  it  cannot 
be  properly  calculated  until  noon,  when  the  latitude  will 
be  determined.  After  that,  the  latitude  can  be  carried 
backwards  by  D.  R.  to  the  moment  of  the  forenoon  time- 
sight,  and  the  latter  can  be  calculated. 

But  if  the  navigator,  because  of  emergency,  needs  his 
longitude  at  once,  after  taking  the  forenoon  time-sight,  he 
must  obtain  the  latitude  by  a  D.  R.  calculation  based  on  the 
last  good  noon-sight.  Most  navigators  calculate  morning 
time-sights  in  this  way,  and  then  repeat  the  calculation 
after  the  new  noon-sight  has  been  obtained.  The  latter 
calculation  will  be  preferable  to  the  former,  because  the 
further  the  latitude  is  carried  along  by  D.  R.,  the  less  accurate 
will  it  be.  And  any  error  in  the  latitude  used  in  the  calcula- 
tion will  impress  a  consequent  error  on  the  calculated  longi- 
tude. 

We  shall  now  work  some  time-sight  examples.  On  board 
ship,  at  sea,  Dec.  18,  1917,  in  the  afternoon,  D.  R.  latitude 
42°  20'  N.,  D.  R.  longitude  35°  16'  W.,  the  altitude  of  sun's 
lower  limb  was  observed  to  be  14°  19'.  The  time  was  taken 
with  the  navigator's  watch,  and  was  2h  29m  58*.  A  com- 
parison of  the  watch  and  ship's  chronometer  gave  C.  —  W.  = 
2h  27m  8*.  The  chronometer  correction  was  2m  8*  slow  of 
G.  M.  T.  The  index  correction  of  the  sextant  was  +  4'  ; 
height  of  eye,  24  ft.  Calculate  the  ship's  longitude. 

We  have  first  to  find,  for  the  moment  of  the  observation, 


102  'NAVIGATION 

values  of  the  declination  and  equation  of  time.     To  do  this, 
we  have : 

Watch  time  of  observation 2A   29m  58«   (1) 

C.  -W 2     27       8    (2) 

Adding  (1)  and  (2)  gives  chronometer  time  of 

observation 4  57  6  (3) 

Chronometer  correction,  slow 2       8     (4) 

Adding  (3)  and  (4)  gives  G.  M.  T.  of  observation  4     59     14    (5) 

For  the  G.  M.  T.  (5)  we  interpolate  the  declina- 
tion (p.  76),  finding -  23°  24'  (6) 

and  for  the  same  G.  M.  T.  we  interpolate  the 

equation  of  time +  3m  21s  (7) 

Now,  adding  (5)  and  (7)  gives  Greenwich  ap- 
parent time  of  observation •  5h  2m  35»  (8) 

Next  we  inspect  the  formulas  (p.  100),  choosing  (2)  be- 
cause latitude  is  +  and  declination  — ,  and  (5)  because  the 
sight  was  an  afternoon  one. 

We  now  have,  from  line  (6),  declination  (disregard- 
ing sign) 23°  24'  (9) 

to  which,  by  formula  (2),  we  add 90       0  (10) 

giving  p 113     24  (11) 

The  observed  altitude  was 14     19  (12) 

Index  correction +4  (13) 

Adding  (12)  and  (13)  gives  corrected  altitude 14     23  (14) 

Correction,  Table  6 +12  (15) 

Correction,  Table  7 : -  5  (16) 

Adding  (14),  (15),  (16)  gives  finally  corrected  altitude  14     30  (17) 

The  latitude  by  D.  R.  is 42     20  (18) 

Adding  (11),  (17),  (18)  gives 170     14  (19) 

Halving  (19)  gives  (by  formula  (3),  p.  100)  s 85       7  (20) 

Subtracting  (17)  from  (20)  gives  (s  -  alt.) 70     37  (21) 

Next  we  apply  formula  (5),  p.  100.     We  have: 

sec  lat.  (18)  from  Table  4,  page  238 0.13121     (22) 

esc  p  (11)  from  Table  4,  page  219 0.03727     (23) 

cos  s  (20)  from  Table  4,  page  200 8.93007     (24) 

sin  (s  -  alt.)  (21)  from  Table  4,  page  215 9.97466     (25) 

sum  (22)  to  (25)  =  hav.  T,  by  formula  (5) 9.07321 l  (26) 

1  This  sum  has  been  diminished  by  10  arbitrarily  (see  p.  25), 
which  must  always  be  done  when  the  sum  of  logs  is  larger  than  10. 


OLDER  NAVIGATION  METHODS  103 

T,1  corresponding  to  (26)  from  Table  10,  page  260,  is  2ft  40W  59*  (27) 
Greenwich  apparent  time  (8)  by  watch  and 

chronometer  is 5  2  35  (28) 

Subtract  (27)  from  (28),  giving  time  difference 

between  ship  and  Greenwich 2  21  36  (29) 

Turning  (29)  into  degrees  with  Table  9,  page  249, 

gives 35°  24'  W.  (30) 

and  (30)  is  the  ship's  longitude  from  this  time-sight. 

Upon  comparing  the  D.  R.  longitude  (35°  16'  W.)  with  the 
result  of  the  time-sight  (35°  24'  W.),  we  find  that  the  ship 
is  8'  west  of  her  D.  R.  position.  This  means,  of  course,  that 
there  has  been  a  westerly  "set"  of  current  in  the  interval 
between  the  last  accurate  determination  of  longitude  and 
the  present  one.  It  would  be  proper  for  the  navigator  to 
calculate  from  this  the  amount  of  westerly  drift  per  hour, 
and  to  allow  for  it  in  carrying  forward  his  longitude  by  D.  R. 
from  the  present  time-sight.  It  is  also  clear  that  the 
northerly  or  southerly  set  of  the  current  can  be  similarly 
measured  and  allowed  for  by  comparing  the  D.  R.  latitude 
with  the  latitude  from  a  noon-sight  (cf.  p.  95).  It  is  the 
general  custom  of  navigators  to  ascribe  such  differences  to 
ocean  currents,  never  to  uncertainty  in  the  astronomic  results. 
Dead  reckoning  is  never  allowed  any  weight  as  against  a 
sextant  observation. 

The  reader  will  have  noticed  that  the  foregoing  calculation 
has  been  made  in  great  detail,  so  that  a  beginner  may  have 
no  difficulty  in  understanding  it.  But  a  practiced  navigator 
would  of  course  work  the  calculation  in  a  much  more  con- 
densed form,  in  such  a  way  as  to  bring  the  logarithms  next 
to  the  numbers  to  which  they  belong.  We  shall  therefore 
now  repeat  the  same  example  in  such  a  condensed  form : 

1  If  the  observation  had  been  made  before  noon,  we  should  have 
used  formula  (4)  and  should  here  have  obtained  24*  — •  T,  instead 
of  T.  This  24*  -  T  would  then  be  subtracted  from  24A,  to  get 
T,  before  continuing  the  calculation.  Thus  the  form  of  calculation 
would  contain  another  line  between  (27)  and  (28),  in  the  case  of 
a  forenoon  observation. 


104 


NAVIGATION 


TIME-SIGHT,  CONDENSED  FORM.    SUN 


Watch  time : 
C.  -  W. : 

Chr.  time : 
Chr.  corr'n : 


29"»  58*  (1) 


2     27 
4     57 

+    2 


G.  M.  T. :  IS'"  4  59 
3 
2 


Eq.  of  time  : 
G.  app.  time  :    5 


8 

6 

8 

14 

21 

35 


(2) 
(3) 
(4) 
(5) 
(7) 
(8) 


Obs'd  alt.  : 
Index  : 
Table  6  : 
Table  7  : 
Corr'd  alt.  : 

14°  19'  (12) 
+    4    (13) 
+  12    (15) 
-    5    (16) 
14  30   (17) 

Decl.  18th,  4A 
H.  D.: 
Decl.  4»  59"»  : 
p: 


23e 

23 
113 


23'.7 
0.1 

24         (6) 
24       (11) 


Eq.  time,  18th,  4* :  +  3"»  22».3 
H.  D.:  1.2 

Eq.  time,  4ft  59TO:    +3      21.1 


(7) 


Corr'd  alt. 
Lat.,  D.  R 
p: 
sum  of  3  : 
s  : 
s  —  alt.  : 

By  chron., 

:         14°  30'  (17) 
.  :        42    20    (18)     sec  lat.  : 
113    24    (11)     esc  p: 

0.13121   (22) 
0.03727  (23) 

8.93007  (24) 
9.97466  (25) 

2)170    14    (19) 

85      7    (20)     cos  s  : 
70    37    (21)     sin  (s-alt.): 
sum  of  4  : 

T  =  ship's  app.  time  : 
Greenwich  app.  time  : 
Longitude  : 
or: 

9.07321  (26)  =  hav.  T 
(or  24*  -  T)  i 
2»  40"»  59*  (27) 
5       2     35      (8) 

2"   21"*  36s   (29) 
35°   24'   W.  (30) 

When  the  object  observed  is  a  star  or  planet,  the  choice 
between  formulas  (4)  and  (5),  p.  100,  is  not  quite  the  same 
as  in  the  case  of  a  solar  time-sight.  We  must  use  (4)  if  there 
is  any  east  in  the  star's  bearing  at  the  moment  of  observation  ; 
and  (5),  if  there  is  west  in  the  bearing.  The  more  nearly  the 
star  bears  due  east  or  west,  the  more  accurate  will  be  the 
resulting  longitude.  The  use  of  formulas  (1),  (2),  and  (3) 
is  the  same  as  for  the  sun ;  but  T,  in  the  case  of  a  star,  is  no 
longer  the  ship's  apparent  solar  time.  Instead,  it  is  called 


1  See  p.  103,  footnote. 


OLDER  NAVIGATION  METHODS  105 

the  star's  " hour-angle."  To  get  the  longitude,  we  must 
first  (p.  85)  calculate  the  Greenwich  sidereal  time  corre- 
sponding to  the  G.  M.  T.  of  the  observation,  as  taken  from 
the  chronometer,  duly  corrected  for  error  and  rate;  and 
then  use  the  following  formulas : 

(6)  Greenwich  sid.  time  —  right-ascension  of  star  =  Greenwich 
hour-angle. 

,_     J  West  long.  =  Greenwich  hour-angle  —  T, 
\  East  long.  =  T  —  Greenwich  hour-angle. 

As  an  example  of  a  star  observation  we  shall  take  the 
following : 

At  sea,  just  before  sunrise,  Dec.  17,  1917,  off  Cape  Agulhas, 
latitude  by  D.  R.  35°  20'  S.,  longitude  by  D.  R.  20°  41'  E., 
the  altitude  of  Sirius  was  measured,  and  found  to  be  40°  3'. 
The  star  bore  west,  and  the  height  of  eye  was  22  ft.  Index 
correction  was  +  5'.  Time  by  watch,  16A  29m  48',  or  4*  29W 
48*  A.M.,  civil  time,  Dec.  18;  C.  -  W.,  -  lh  23OT  50";  chro- 
nometer fast  of  G.  M.  T.  2m  28s. 

The  calculation  would  proceed  thus : 

Watch  time  of  observation 16*   29™  48«     (1) 

C.  -  W -  1    23     50      (2) 

Adding  (1)  and  (2),  having  regard  to  —  sign  of  (2), 

gives  chronometer  time  of  observation 15  5  58  (3) 

Chronometer  correction,  fast —  2     28      (4) 

Adding  (3)  and  (4),  having  regard  to  —  sign  of  (4), 

gives  G.  M.  T.  of  observation 15  3  30  (5) 

Right  ascension  mean  sun,  Greenwich  mean  noon, 

Dec.  17  (p.  83) 17  42  10  (6) 

Correction  for  "  time  past  noon  "  (see  p.  84) ....  2     28      (7) 

Adding  (6)  and  (7)  gives  right  ascension  of  mean 

sun 17  44  38  (8) 

Adding  (5)  and  (8)  (see  p.  85)  gives  Greenwich 

sidereal  time  of  the  observation 8 l  48  8  (9) 

Right  ascension  of  Sirius,  Dec.  17,  is  (p.  91) ....     6     41     34    (10) 

Subtracting  (10)  from  (9)  gives  Greenwich  hour- 
angle  (formula  (6),  above) 2  6  34  (11) 

1  This  is  really  32* ;   but  24*  is  dropped  arbitrarily. 


106  NAVIGATION 

Next  we  calculate  T  by  formula  (5),  p.  100.    We  have : 

Declination  of  Sirius,  Dec.  17  (p.  92)  -  16°  36'  (12) 

By  formula  (1),  p.  100,  subtract  (12)  from  90°, 

without  attention  to  sign  of  (12), giving  p.  .  73  24  (13) 

The  observed  altitude  was 40  3  (14) 

The  index  correction  was +5  (15) 

Table  6  correction -  1  (16) 

Table  7  correction —  5  (17) 

Adding  (14),  (15),  (16),  (17),  having  regard  to 

signs,  gives  corrected  altitude 40  2  (18) 

The  latitude  by  D.  R.  was 35  20  (19) 

Adding  (13),  (18),  and  (19)  gives 148  46  (20) 

Halving  (20)  gives  s 74  23  (21) 

Subtracting  (18)  from  (21)  gives  (s  -  altitude)  .  .  34  21  (22) 

Now  applying  formula  (5),  page  100,  we  have  : 

sec  latitude  (19)  from  Table  4,  page  231   0.08842       (23) 

esc  p  (13)  from  Table  4,  page  212 0.01849       (24) 

cos  s  (21)  from  Table  4,  page  211 9.43008       (25) 

sin  (s  -  altitude)  (22)  from  Table  4,  page  230 9.75147       (26) 

Summing  (23)  to  (26)  gives  hav.  T7,  by  form.  (5)  .  .  9.28846  *     (27) 
T 2 corresponding  to  (27),  from  Tab.  10,  p.  263  is .  .  3*  29"  14«        (28) 
Difference  between  (28)  and  (11)  is  the  longi- 
tude by  formula  (7),  page  105 1    22     40  E.   (29) 

Turning  (29)  into  degrees  with  Table  9,  page 

249,  gives 20°  40'  E.   (30) 

The  D.  R.  longitude,  20°  41'  E.,  was  therefore  within  1'  of 
the  longitude  from  this  time-sight,  and  this  shows  that  the 
ship  has  not  been  affected  by  ocean  currents  since  the  last 
observation.  It  is  also  interesting  to  note  how  near  sunrise 
the  observation  was  made.  The  twilight  must  have  been 
quite  strong,  and  the  star  therefore  dim.  But  star  observa- 
tions can  be  made  best  in  twilight  because  the  horizon  line 
can  then  be  seen  distinctly. 

1  This  sum  has  also  been  diminished  by  10  (see  footnote,  p.  102). 

2  Might  be  24*  -  T,  if  the  star  bore  E.  instead  of  W.  (see  footnote, 
p.  103). 


OLDER  NAVIGATION  METHODS 


107 


The  foregoing  example  can  of  course  also  be  arranged  in 
condensed  form,  as  follows : 

TIME-SIGHT,   CONDENSED   FORM.    STAR 

Watch  time : 

C.  -  W. : 

Chr.  time : 

Chr.  corr'n : 

G.  M.  T. : 

R.  A.  mean  sun : 

Corr'n,  past  noon : 

Greenw'h  sid.  time : 

R.  A.  of  Sirius : 

Greenwich  hour-ang. 

T7.,  from  (27) : 

Long.: 

or: 


R.  A.  of  Sirius : 

Dec.  of  Sirius : 

p: 

sec  lat. : 

esc.  p : 

cos  s: 

sin  (s-alt.) : 

sum  of  4 : 


16* 

29" 

48-         (1) 

Obs'd 

alt.: 

40° 

3' 

(14) 

-1 

23 

50          (2) 

Index 

: 

+  5 

(15) 

15 

5 

58          (3) 

Table 

6: 

-1 

(16) 

-2 

28          (4) 

Table 

7: 

-5 

(17) 

15 

3 

30          (5) 

Corr'd  alt.  : 

40 

2 

(18) 

17 

42 

10          (6) 

Lat.  D.  R.  : 

35 

20 

(19) 

2 

28          (7) 

p: 

73 

24 

(13) 

8 

48 

8          (9) 

sum: 

2)148 

46 

(20) 

6 

41 

34        (10) 

s  : 

74 

23 

(21) 

:  2 

6 

34        (11) 

(s-alt.): 

34 

21 

(22) 

3 

29 

14        (28) 

1 

22 

40  E.  (29) 

20° 

40'  E.  (30) 

6*  41"  34' 

(10) 

-  16°   36' 

(12) 

73     24 

(13) 

0.08842 

(23) 

0.01849 

(24) 

9.43008 

(25) 

9.75147 

(26) 

9.28846  (27)  =  hav.  T  (or  24*  -  T) 


Having  now  fully  explained  both  the  noon-sight  and  the 
time-sight,  we  shall  close  this  chapter  with  a  strong  recom- 
mendation to  young  navigators  to  familiarize  themselves  with 
the  observation  of  stars.  These  always  furnish  a  valuable 
check  on  sun  observations  :  and  at  times  of  danger  may  save 
the  ship  when  clouds  have  obscured  the  sun  for  days,  and 
clearing  occurs  after  sunset.  It  is  easy  to  learn  to  know  the 
principal  stars  from  Jacoby's  "  Astronomy/'  Chapter  III, 
"How  to  Know  the  Stars." 


See  footnote,  p.  103. 


CHAPTER  IX 
NEWER  NAVIGATION  METHODS 

THE  reader  may  have  noticed  in  Chapter  VIII  that  there 
is  a  very  definite  difference  between  the  determination  of 
latitude  by  a  noon-sight  and  longitude  by  a  time-sight :  for 
the  latitude  is  obtained  without  previous  knowledge  of  the 
longitude;  but  to  get  the  longitude,  a  previous  knowledge 
of  the  latitude  is  essential.  This  is,  of  course,  a  decided 
disadvantage  in  determining  longitude,  nor  is  there  any 
practicable  direct  way  to  get  the  longitude  without  first 
knowing  the  latitude. 

We  have  also  seen  (p.  101)  that  any  existing  uncertainty 
in  our  knowledge  of  the  latitude  will  produce  an  error  in  the 
longitude  computed  from  a  time-sight.  In  situations  of 
danger  it  is  important  to  ascertain  how  great  this  longitude 
error  may  be.  Suppose,  for  instance,  we  have  calculated 
a  time-sight  with  a  D.  R.  latitude  that  we  suspect  may  be 
as  much  as  10'  too  small ;  and  we  wish  to  know  how  much 
our  computed  longitude  may  have  been  thereby  put  wrong. 
The  obvious  way  to  find  out  is  to  recompute  the  longitude 
with  an  assumed  latitude  10'  larger  than  the  D.  R.  latitude. 
The  resulting  longitude  will  then  show  the  extreme  range 
of  error  that  must  have  been  produced  if  the  D.  R.  latitude 
was  10'  too  small. 

A  third  calculation,  with  an  assumed  latitude  10'  smaller 
than  the  D.  R.  latitude,  will  similarly  exhibit  the  extreme 
possible  range  of  longitude  error  in  the  other  direction. 
Thus  these  two  extra  calculations  will  show  the  limits  of 
longitude  error  that  might  be  caused  by  a  range  of  20'  in 
the  possible  error  of  the  D.  R.  latitude. 

108 


NEWER  NAVIGATION  METHODS  109 

This  rather  obvious  procedure  was  probably  used  long 
ago  by  more  than  one  intelligent  navigator ;  but  it  was  first 
published  in  1837  by  Thomas  H.  Sumner,  an  American 
merchant  captain.  He  used  the  method  in  dramatic  cir- 
cumstances of  great  danger ;  and  he  brought  his  ship  safely 
into  port.  According  to  his  own  account,  he  made  three 
calculations  of  the  longitude,  using  three  assumed  latitudes 
differing  by  10',  and  he  of  course  obtained  three  different 
longitudes.  He  then  marked  or  plotted  (p.  55)  on  his  chart 
the  point  indicated  by  the  first  assumed  latitude  and  its 
computed  longitude.  At  this  point  the  ship  must  have  been 
located,  if  the  first  assumed  latitude  had  been  correct.  The 
other  two  latitudes,  with  their  computed  longitudes,  indicated 
two  more  points  on  the  chart ;  and  at  one  of  these  points  the 
ship  must  have  been,  if  either  of  these  additional  latitudes 
was  correct. 

Sumner  found  that  the  three  points  on  the  chart  lay  in  a 
straight  line;  and  it  became  at  once  evident  that  whatever 
latitude  he  might  assume  (within  reason)  he  would  always 
get  a  point  on  the  same  straight  line,  after  computing  the 
longitude.  In  other  words,  although  he  did  not  know  his 
latitude  accurately,  and  so  could  not  compute  his  longitude 
accurately,  yet  he  had  found  a  straight  line  on  the  chart 
upon  which  his  ship  was  surely  situated. 

Such  a  line  can  always  be  found  in  the  way  Sumner  found 
it,  or  in  some  preferable  modern  way;  and  such  a  line  we 
shall  call  a  "  Sumner  line,"  though  some  writers  on  naviga- 
tion prefer  to  call  it  a  "line  of  position." 

On  the  occasion  of  laying  down  his  line,  Sumner  found  that 
it  passed  directly  through  Small's  Light,  near  the  Irish  coast ; 
and  as  the  line  bore  E.N.E.  on  his  chart,  he  simply  put 
the  ship  on  that  course,  and  in  less  than  an  hour  he  "made" 
Small's  Light,  actually  bearing  E.N.E.  J  E.,  and,  as  he  says, 
"close  aboard."  He  had  had  no  observations  after  passing 
longitude  21°  W.,  until  the  morning  of  Dec.  17,  when  these 
historic  events  occurred.  He  was  off  a  rocky  lee  shore,  in 


110  NAVIGATION 

the  midst  of  a  winter  gale,  after  crossing  the  Atlantic ;  only 
a  seaman  can  understand  the  relief  he  must  have  felt  when 
that  light  suddenly  appeared  off  the  bow. 

We  have  given  this  account  of  Summer's  experience  to 
impress  on  the  young  navigator  that  he  must  positively 
familiarize  himself  with  the  Sumner  method  of  navigation. 
Should  we  be  so  fortunate  as  to  have  any  experienced  navi- 
gator among  our  readers,  we  ask  him  to  try  the  Sumner 
method  once  more,  in  the  manner  explained  below,  even  if 
he  may  have  found  it  troublesome  in  the  past  on  account  of 
certain  difficulties  in  its  application.  For  the  Sumner 
method  is  the  best  method  of  navigation  on  all  oceans  and 
at  all  times  :  even  when  a  noon-sight  is  available  for  latitude, 
it  is  better  to  treat  it  as  a  Sumner  observation,  and  work 
out  the  Sumner  line. 

The  principal  objection  urged  against  it  by  certain  prac- 
tical navigators  arises  from  the  small  scale  of  existing  ocean 
track  charts,  on  which  a  distance  of  10'  is  represented  by 
about  -J  inch.  A  line  like  Sumner's,  20'  long,  would  have 
only  a  length  of  J  inch  on  the  chart ;  and  such  a  little  line 
would  not  be  long  enough  to  show  accurately  the  direction 
in  which  it  pointed.  When  near  a  coast,  as  in  Sumner 's 
case,  this  difficulty  disappears,  because  navigators  always 
have  (or  always  should  have  and  use)  the  large  scale  charts 
that  can  be  obtained  for  coastwise  waters. 

But  it  is  inconvenient  for  navigators  to  begin  using  a 
method  off  the  coast,  on  the  last  day  of  a  voyage,  different 
from  the  form  employed  for  many  days  at  sea.  Therefore, 
some  authorities  recommend  the  construction  of  a  special 
large  scale  chart,  with  its  latitude  and  longitude  lines,  each 
time  an  observation  is  made  throughout  the  voyage,  so  that 
the  Sumner  line  can  always  be  drawn  on  a  sufficiently  large 
scale.  It  is  no  wonder  that  navigators  have  not  generally 
adopted  this  somewhat  laborious  proceeding;  and  in  the 
method  given  below  we  shall  utilize  the  Sumner  idea  without 
requiring  any  lines  to  be  drawn  on  charts. 


NEWER  NAVIGATION  METHODS  111 

Another  objection  to  Sumner  navigation  is  that  it  requires 
too  much  calculation ;  three  longitude  calculations  for  one 
observation,  as  Sumner  practiced  it.  This  objection  is  also 
quite  removed  now  by  the  use  of  suitable  tables  such  as  we 
give  in  the  present  volume. 

But  before  proceeding  to  explain  these  tables,  we  must 
outline  briefly  the  real  principle  on  which  rests  the  com- 
plete utilization  of  the  Sumner  method  on  the  open  sea. 
There  the  navigator  wants  to  know  the  ship's  position  in 
both  latitude  and  longitude ;  and  will  not  be  satisfied  with 
a  mere  line,  with  the  ship  " somewhere  on  the  line."  Along 
the  coast  such  a  line  might  help  him  to  find  Small's  Light  ; 
but  he  is  not  looking  for  coast  lights  at  sea. 

And  the  Sumner  method  takes  care  of  this  matter  in  the 
simplest  possible  way.  We  have  seen  (p.  88)  that  two 
different  observations  are  always  necessary  by  any  method 
to  get  both  latitude  and  longitude.  But  two  such  observa- 
tions by  the  Sumner  method  give  two  different  lines  on  the 
chart :  and  as  the  ship  must  be  located  on  both  lines,  her 
actual  position  must  be  at  their  point  of  intersection.  We 
shall  show  how  the  required  latitude  and  longitude  of  the 
ship  at  the  point  of  intersection  can  be  found  by  a  simple 
calculation,  without  the  drawing  of  any  lines  on  the  chart. 

Coming  now  to  the  modern  method  of  calculating  a  Sum- 
ner line,  we  must  first  state  a  general  fundamental  principle 
that  may  be  easily  verified  by  geometrical  considerations. 
The  true  bearing  (p.  44)  of  a  Sumner  line  on  a  chart  is 
always  90°  greater  than  the  true  bearing  or  azimuth  (p.  44) 
of  the  sun  (or  star)  at  the  moment  of  observation.  Or,  in 
other  words,  the  Sumner  line  bears  at  right  angles  to  the 
sun  at  the  time  of  observation. 

We  shall  show  how  the  bearing  or  azimuth  of  the  sun  can 
always  be  found  from  suitable  " azimuth  tables";  but  the 
Sumner  line  is  not  completely  known  from  its  bearing  alone. 
To  locate  it  properly  it  is  necessary  to  know  in  addition  the 
latitude  and  longitude  of  some  point  on  the  line,  which  we 


112  NAVIGATION 

will  call  a  "Sumner  point."  Then,  knowing  such  a  point  of 
the  line,  and  the  bearing  of  the  line,  we  may  say  we  know  the 
line  completely,  and,  if  necessary,  could  draw  it  on  a  chart. 

Now  to  find  the  required  Sumner  point.  We  always  have 
the  D.  R.  position  of  the  ship  at  the  moment  of  observation ; 
which  we  will  call  the  "D.  R.  point."  It  is  easy  to  find 
out  if  the  D.  R.  point  is  also  a  Sumner  point.  It  is  merely 
necessary  to  calculate  what  the  sun's  altitude  would  be  for 
a  ship  at  the  D.  R.  point,  and  then  compare  this  calculated 
altitude  with  the  one  actually  observed.  If  the  D.  R.  point 
was  really  a  Sumner  point  (which  will  rarely  happen),  the 
two.  altitudes  will  agree ;  if  not,  the  amount  of  disagreement 
will  show  how  far  the  D.  R.  point  is  distant  from  the  nearest 
Sumner  point.1 

The  first  step,  then,  in  Sumner  navigation,  is  the  calcula- 
tion of  the  altitude,  supposing  the  ship  to  be  at  the  D.  R. 
point  at  the  moment  of  observation.  To  do  this  for  a  sun 
observation,  we  first  calculate  the  Greenwich  apparent  time 
(abbreviated  G.  A.  T.)  of  the  observation,  just  as  was  done 
in  the  case  of  a  time-sight  on  p.  102.  To  this  G.  A.  T.  we 
then  add  the  ship's  D.  R.  longitude,  if  east,  or  subtract  it,  if 
west,  to  get  T  (p.  100),  the  ship's  apparent  time  of  the  ob- 
servation. We  then  use  the  formulas  on  p.  113,  in  which 
X  and  Z  are  " auxiliary  angles"  required  in  the  calculations, 
but  not  otherwise  of  special  interest.  These  formulas  are 
called  the  "  cosine-haversine  "  formulas. 

There  are  several  other  sets  of  formulas  with  which  the 
same  problem  can  be  solved.  One  set,  called  the  "  haversine  " 
formulas,  involves  the  use  of  haversines  only;  another, 
called  the  "  sine-cosine "  formulas,  solves  the  problem  with 
sines  and  cosines.  But  neither  is  preferable  to  the  following 
cosine-haversine  set. 

1  This  method  is  often  called  the  Marcq  Saint  Hilaire  method ; 
but  it  should  probably  be  credited  to  Lord  Kelvin,  who  published 
"  Tables  for  Facilitating  Sumner's  Method  at  Sea  "  in  1876.  These 
tables  follow  the  method  described  above. 


NEWER  NAVIGATION  METHODS  113 

If  observation  was  made  before  noon,  ship's  time, 

hav.  X  =  cos  lat.  +  cos  dec.  +  hav.  (24*  -  T) ,  (1 ) 

If  observation  was  made  after  noon,  ship's  time, 

hav.  X  =  cos  lat.  +  cos  dec.  +  hav.  T,  (2) 

lat.  —  dec.  =  diff.1  of  lat.  and  dec.,  if  both  are  +  or  both  — ,  (3) 

lat.  —  dec.  =  sum1  of  lat.  and  dec.  if  one  is  +  and  one  — ,  (4) 

No.  hav.  Z  =  No.  hav.  (lat.  -  dec.)  +  No.  hav.  X,  (5) 

Alt.  =  90°  -  Z.  (6) 

Now  we  can  compare  the  altitude  computed  by  formula 
(6)  with  the  observed  altitude,  fully  corrected  for  index 
error,  etc.  The  difference  between  the  two  altitudes  in 
minutes  will  be  the  distance  in  miles  of  the  nearest  Sumner 
point  from  the  D.  R.  point,  for  the  minute  and  nautical 
mile  here  correspond,  as  they  do  in  the  case  of  differences  of 
latitude  (p.  15).  The  bearing  of  the  Sumner  point  from  the 
D.  R.  point  will  be  the  same  as  the  sun's  azimuth  if  the  ob- 
served altitude  is  greater  than  the  computed  altitude  :  but  if 
the  observed  altitude  is  less  than  the  computed,  the  bearing  of 
the  Sumner  point  will  be  180°  greater  than  the  sun's  azimuth. 

The  bearing  and  distance  of  the  Sumner  point  from  the 
D.  R.  point  once  known,  it  is  easy,  by  means  of  the  traverse 
table  (p.  10),  to  obtain  the  latitude  and  longitude  of  the 
Sumner  point  from  the  known  latitude  and  longitude  of 
the  D.  R.  point ;  or,  which  is  the  same  thing,  from  the  ship's 
D.  R.  latitude  and  longitude. 

Before  giving  examples  of  these  calculations,  it  remains 
to  show  how  the  sun's  bearing  or  azimuth  can  be  taken  from 
Table  11  (p.  284),  called  the  azimuth  table.  The  pair  of 
arguments  (p.  11)  for  entering  this  table  are:  first,  in  the 
left-hand  column,  the  declination,  which  is  here  used  without 
regard  to  its  sign;  and  second,  in  the  four  topmost  hori- 

1  In  using  formulas  (3)  and  (4),  pay  no  attention  to  +  or  — 
signs  after  the  right  formula  is  once  chosen.  The  difference  between 
latitude  and  declination  is  always  taken  by  subtracting  the  smaller 
from  the  larger ;  and  the  sum  by  adding  them,  without  regarding 
their  +  or  —  signs.  Cf.  also  p.  89. 


114  NAVIGATION 

zontal  lines,  T  (p.  100),  the  ship's  apparent  time  at  the 
moment  of  observation. 

Having  found  this  pair  of  arguments,  we  look  in  the 
column  under  T,  and  in  the  horizontal  line  opposite  the 
declination.  There  we  find  an  "index  number."  Next  we 
look  up  the  altitude,  as  computed  by  formula  (6),  page  113, 
in  the  right-hand  column  of  the  azimuth  table,  and  follow 
along  the  horizontal  line  belonging  to  that  altitude,  until 
we  reach  a  number  equal  (or  nearly  equal)  to  the  index 
number.  Then  we  go  down  the  column  containing  this 
second  appearance  of  the  index  number,  and  find  the  azi- 
muth at  the  bottom  of  the  page.  The  table  gives  approxi- 
mate azimuths  only,  but  the  approximation  is  sufficient  for 
our  present  purpose. 

The  azimuths  at  the  bottom  of  the  page  appear  in  four 
horizontal  lines,  of  which  the  upper  two  belong  to  forenoon 
observations,  and  the  lower  two  to  afternoon  observations. 
All  azimuths  are  counted  from  the  north,  through  east, 
south,  and  west,  from  0°  to  360°,  like  compass  courses  in 
United  States  Navy  practice  (p.  41).  It  is  important  for 
the  navigator  to  record,  at  the  time  of  observation,  the  word 
" forenoon"  or  " afternoon,"  and  also  the  sun's  roughly 
approximate  bearing,  to  aid  in  choosing  which  of  the  azi- 
muths at  the  bottom  of  the  tabular  page  is  the  right  one. 
The  record  showing  whether  the  observation  was  made  in 
the  forenoon  or  afternoon  limits  the  choice  to  two  of  the  lines 
of  azimuths;  and  if  there  is  any  doubt  remaining  between 
these  two,  the  following  rules  may  clear  it  up. 

When  latitude  is  +  and  declination  — ,  azimuth  is  between 
90°  and  270°; 

When  latitude  is  -f  and  declination  +,  if  declination  is 
greater  than  latitude,  azimuth  is  not  between  90°  and  270° ; 

When  latitude  is  —  and  declination  — ,  if  declination  is 
greater  than  latitude,  azimuth  is  between  90°  and  270° ; 

When  latitude  is  —  and  declination  -f-,  azimuth  is  not 
between  90°  and  270°. 


NEWER  NAVIGATION  METHODS  115 

In  other  cases,  and  especially  when  latitude  and  declina- 
tion are  nearly  equal,  the  foregoing  rules  are  insufficient,  and 
we  must  consult  Table  12  (p.  290),  the  "  auxiliary  azimuth 
table."  This  table  has  latitude  and  declination  for  its  pair 
of  arguments,  the  former  in  the  left-hand  vertical  column, 
the  latter  in  the  topmost  horizontal  line :  and  in  using  the 
table  it  is  not  necessary  to  pay  attention  to  the  +  and  - 
signs  of  latitude  and  declination.  Start  with  the  latitude, 
and  follow  its  horizontal  line  to  the  right  until  you  reach  the 
column  having  the  declination  at  its  head.  There  you  will 
find  an  "auxiliary  angle,"  which  must  be  compared  with 
the  altitude  computed  by  formula  (6),  page  113.  Then : 

If  the  computed  altitude  is  greater  than  the  auxiliary 
angle,  and  if  latitude  is  +,  azimuth  is  between  90°  and  270° ; 

If  the  computed  altitude  is  less  than  the  auxiliary  angle, 
and  if  latitude  is  -,  azimuth  is  between  90°  and  270° ; 

If  the  computed  altitude  is  less  than  the  auxiliary  angle, 
and  if  latitude  is  +,  azimuth  is  not  between  90°  and  270° ; 

If  the  computed  altitude  is  greater  than  the  auxiliary 
angle,  and  if  latitude  is  — ,  azimuth  is  not  between  90°  and 
270°. 

It  will  rarely  happen  that  any  of  the  foregoing  rules  will 
be  needed,  if  the  navigator  will  make  a  careful  observation 
of  the  sun's  azimuth  with  the  azimuth  circle  or  pelorus 
(p.  44),  as  soon  as  possible  after  the  sextant  altitude  has 
been  observed.  The  ship's  course  should  also  be  specially 
recorded  when  this  observation  is  made.  This  proceeding 
is  not  merely  a  convenience  to  avoid  consulting  the  fore- 
going rules  in  using  the  azimuth  table :  it  is  really  essential 
to  safe  navigation,  for  a  comparison  of  the  observed  azi- 
muth with  that  derived  from  the  table  will  make  the  com- 
pass error  (p.  43)  known.  The  variation  is  known  from  the 
chart ;  so  that  if  we  observe  the  compass  error,  we  can  allow 
for  the  variation,  and  get  the  deviation.  This  can  then  be 
compared  with  the  deviation  table  (p.  48),  to  see  if  there  has 
been  any  change  in  the  compass  since  leaving  port.  It  is 


116  NAVIGATION 

a  great  advantage  of  the  Sumner  method  that  the  sun's 
azimuth  comes  out  as  a  sort  of  by-product,  so  that  the  com- 
pass can  be  verified  without  any  additional  special  calcu- 
lations. 

We  shall  now  illustrate  all  the  above  considerations  by 
means  of  examples ;  beginning  with  the  observation  already 
treated  as  a  time-sight  (p.  101).  That  observation  we  shall 
now  work  by  the  Sumner  method.  From  page  101  we  take 
the  following : 

Date  of  observation,  Dec.  18,  1917,  in  the  afternoon;  D.  R. 
latitude,  42°  20'  N. ;  D.  R.  longitude,  35°  16'  W. ;  altitude  observed, 
14°  19' ;  time  by  watch,  2*  29W  58* ;  C.  -  W.,  2*  27m  8« ;  chronometer 
correction,  2m  8*  slow  of  G.  M.  T. ;  index  correction,  +  4' ;  height  of 
eye,  24  ft. 

From  the  preparatory  part  of  the  calculation  (p.  102), 
we  also  copy  the  following  additional  numbers : 

Declination,  line  (6),  page  102 -23°  24'     (1) 

Greenwich  apparent  time  (G.  A.  T.)  of  observation, 

line  (8),  page  102 5*  2"  35*     (2) 

We  have  next  to  calculate,  by  the  formulas  on  page  113,  the 
altitude  corresponding  to  the  D.  R.  point,  for  which  the 
latitude  and  longitude  are  given  above.  The  longitude  is 
35°  16'  W.,  or,  at  15°  to  the  hour  (Table  9,  p.  249) : 

D.  R.  longitude  is 2h  2lm  4*  W.  (3) 

Subtracting  (3)  from  (2),  according  to  page  112, 

gives  ship's  apparent  time  of  observation,  T.  .    2    41    31       (4) 

We  are  now  prepared  to  apply  formulas  (1)  to  (6), 
page  113.  We  choose  formula  (2)  for  an  afternoon  obser- 
vation l ;  and  write  : 

1  For  a  forenoon  observation  we  should  choose  formula  (1),  and 
should  therefore  need  to  know  24*  -  T  instead  of  T.  This  would 
make  necessary  another  line  in  the  form  of  calculation,  and  it  would 
follow  line  (4).  This  new  line  might  be  numbered  (4') ;  and  in  it 
would  be  written  24*  -  T,  obtained  by  subtracting  T  (line  4)  from 
24*. 


NEWER  NAVIGATION  METHODS  117 

Cos  lat.,  42°  20'  N.  by  D.  R.  (see  Table  4,  p.  238) ....  9.86879  (5) 

Cos  dec.,  23°  24',  line  (1)  (see  Table  4,  p.  219) 9.96273  (6) 

Hav.  T,  2*  41*  31',  line  (4)  (see  Table  10,  p.  260) ....  9.07596  (7) 

Adding  (5)  to  (7)  gives  hav.  X  (dropping  20,  p.  25) .  .  8.90748  (8) 

Now  we  choose  formula  (4),  because  latitude  and  declina- 
tion are  +  and  —  ; 

The  latitude  is,  by  D.  R. .  .  . 42°  20'    (9) 

Adding   (1)   and   (9)  according  to  formula  (4)  gives 

(lat.  -  dec.) 65°  44'  (10) 

Now  we  have,  Table  10,  page  266,  No.  hav.  of  (10) . .  0.29451   (11) 

No.  hav.  X,1  line  (8) 0.08082  (12) 

Adding  (11)  and  (12),  according  to  formula  (5),  page 

113,  gives  No.  hav.  Z 0.37533  (13) 

,     And  Z,  corresponding  to  (13)  is  found  from  Table  10, 

page  268     75°  34'  (14) 

Then,  by  formula  (6)  computed  altitude  =90°  -  Z  (14), 

or 14°  26'  (15) 

This  computed  altitude  (15)  must  now  be  compared  with 
the  observed  altitude,  fully  corrected.  We  find  : 

Obs'd  alt.,  fully  corrected,  line  (17),  page  102,  is 14°  30'  (16) 

Difference  between  (15)  and  (16),  in  minutes,  is  the 
distance  of  Sumner  point  from  D.  R.  point  in 
miles  (p.  113).  It  is 4  miles  (17) 

Next  we  must  find  the  sun's  azimuth  from  Table  11,  page 
286.  The  top  argument  for  entering  the  table  is  T,  line 
(4),  and  it  must  be  found  in  the  " afternoon"  lines.  The 
argument  for  the  left-hand  column  is  the  declination,  line  (1). 
Under  T,  and  opposite  declination,  we  find  the  tabular  index 
number  5872.2  Then  we  find  the  computed  altitude,  line 
(15),  in  the  right-hand  column  of  Table  11,  page  286,  arid 

1  This  No.  hav.  X  comes  from  Table  10,  page  258,  without  looking 
up  the  angle  X  at  all.     We  simply  find  hav.  X  in  the  table,  and  take 
the  No.  hav.  X  out  of  the  adjoining  heavy  type  column.     No  inter- 
polations are  needed,  the  nearest  tabular  numbers  being  sufficiently 
accurate. 

2  The  index  numbers  and  the  azimuth  need  not  be  very  accurate  : 
it  is  sufficient  to  use  the  nearest  tabular  arguments,  so  that  inter- 
polation is  not  essential. 


118  NAVIGATION 

follow  its  horizontal  line  till  we  again  come  upon  the  index 
number  5872.  It  lies  about  halfway  between  5703  and 
5973.  Going  down  the  two  columns  containing  these  index 
numbers,  we  find  in  the  afternoon  azimuth  lines  two  values 
of  the  azimuth,  217°  and  323°.  The  choice  between  these 
two  numbers  would  be  very  easy,  if  the  observer's  record 
contained  even  a  rough  estimate  of  the  sun's  bearing  at  the 
time  of  observation.  We  have  purposely  not  made  this  avail- 
able, so  as  to  show  how  to  consult  the  directions  on  page 
114,  and  there  we  find  that  when  the  latitude  is  +  and  the 
declination  — ,  the  azimuth  is  between  90°  and  270°.  So 
we  finally  choose  217°  for  the  sun's  azimuth. 

Since  the  observed  altitude  (16)  is  greater  than  the  com- 
puted altitude  (15),  the  bearing  of  the  Sumner  point  from 
the  D.  R.  point,  according  to  page  113,  is  the  same  as  the  sun's 
azimuth,  or  217°.  And  as  we  now  know  the  bearing  and 
distance  of  the  Sumner  point  from  the  D.  R.  point,  we  can 
find  its  latitude  and  longitude  by  a  simple  application  of  the 
traverse  table  (p.  154). 

We  have  merely  to  consider  the  bearing  and  distance  to 
be  a  course  angle  and  distance,  and  imagine  a  ship  to  have 
sailed  from  the  one  point  to  the  other.  In  the  present  case, 
the  distance  is  4  miles  (line  17),  the  course  217°  :  and  Table  1 
(p.  164)  gives  the  corresponding  latitude  3'. 2,  departure  2.4. 
The  longitude  difference  is  obtained  from  the  departure  by 
Table  2  (p.  174)  and  is,  for  latitude  42°,  about  3'.2.  Drop- 
ping odd  fractions,  the  latitude  difference  and  longitude  differ- 
ence both  come  out  3'.  The  Sumner  point  is  therefore  3'  dis- 
tant from  the  D.  R.  point  in  both  latitude  and  longitude. 
And  since  the  bearing  217°  indicates  on  the  compass  card 
that  the  Sumner  point  is  south  and  west  of  the  D.  R.  point, 
it  follows  that : 

Lat.  of  Sumner  point  =  D.  R.  lat.  -  3'  = 

42°  20'  N.  (line  9)  -  3' 42°  17'  N.  (18) 

Long,  of  Sumner  point  =  D.  R.  long.  +3' 35   19  W.  (19) 

Azimuth  of  Sumner  line  (p.  Ill) 307°               (20) 


NEWER  NAVIGATION  METHODS 


119 


It  is  important  for  the  reader  to  understand  that  the  fore- 
going calculation  is  given  in  extended  detail  so  as  to  make 
it  easy  for  the  beginner  to  follow.  In  condensed  form, 
we  should  have  the  following  arrangement  of  the  calculation, 
corresponding  to  the  condensed  time-sight  form  (p.  104). 
Part  of  the  work  here  repeated  from  page  104  has  no  attached 
reference  numbers  in  parentheses :  the  new  part  of  the  work 
has  references  to  the  detailed  calculation  just  given. 


SUMNER  LINE,  CONDENSED  FORM.    SUN 


Obs'd  alt. :  14°  19' 

Index:  +    4 

Table  6 :  +12 

Table  7:  -    5 

Corr'dalt.:  14°  30' 


Decl.4*:         23°  23'. 7  S. 

H.  D.:  0.1 

Decl.  4*  59»» :  23°  24'  S. 
Eq.  time,  4* :  +  3»  22«.3 

H.  D.:  1.2 

Eq.  time,  4*  59"* :    +3     21.1 


Watch  time  : 

2*  29«  58« 

C.  -  W.: 

2   27      8 

Chr.  time  : 

4   57      6 

Chr.  corr'n  :             + 

2      8 

G.  M.  T.  18th  : 

4   59    14 

Eq.  of  time  :            -f 

3    21 

G.  app.  time  : 

5     2    35 

D.  R.  long.  : 

2    21      4W.    (3) 

Ship's  app.  time,   T  : 

2   41    31 

(4) 

hav.  T  (or  24^  -T7) 

»:  9.07596 

D.  R.  lat.  : 

42°  20'  N. 

(9) 

cos  lat.  : 

9.86879 

Dec.: 

23   24   S. 

(D 

cos  dec.  : 

9.96273 

sum  =  hav.  X  : 

8.90748 

No.  hav.  X  : 

0.08082  (12) 

No.  hav.  (lat. 

Lat.  -  Dec.  : 

65    44 

(10) 

-  dec.)  : 

0.29451  (11) 

Z: 

75  34 

(14) 

No.  hav.  Z 

0.37533  (13) 

Comp'd  alt.  : 

14   26 

(15) 

Obs'd  alt.  : 

14   30 

(16) 

Diff.  : 

4 

(17) 

Index  No.  : 

5872 

Azimuth  : 

217° 

Lat.  diff  .  : 

3'.2 

Dep.  : 

2.4 

Long.  diff.  : 

3'.2 

D.  R.  lat.  : 

42°  20'  N. 

(9) 

D.  R.  long.  : 

35°  16'  W.     (3) 

Sumner  pt.  lat.  : 

42    17   N. 

(18) 

Sumner  pt.  long.  : 

35    19  W.  (19) 

Azimuth  of  Sumner 

line:  307° 

(20) 

See  footnote,  p.  116. 


120  NAVIGATION 

When  the  object  observed  is  a  star  (cf.  p.  104)  or  planet, 
the  choice  between  formulas  (1)  and  (2),  page  113,  is  not  quite 
the  same  as  in  the  case  of  a  solar  observation.  We  must 
use  formula  (1)  if  the  star  was  on  the  east  side  of  the  sky 
when  observed,  which  might  be  called  a  " forenoon"  observa- 
tion of  the  star ;  and  we  must  use  (2)  if  the  star  was  on  the 
west  side  of  the  sky,  giving  an  " afternoon"  star  observa- 
tion. The  use  of  the  remaining  formulas  (3)  to  (6)  is  the 
same  as  for  the  sun ;  but  T  is  now  no  longer  the  ship's  appar- 
ent time.  Instead,  it  is  the  star's  hour-angle  (p.  104) ; 
to  find  it  for  use  in  formulas  (1)  and  (2),  and  in  Table  11, 
we  must  first  calculate  (p.  85)  the  Greenwich  sidereal 
time  corresponding  to  the  G.  M.  T.  of  the  observation,  as 
taken  from  the  chronometer,  duly  corrected  for  error  and 
rate ;  and  then  use  the  following  formulas  : 

(7)  Greenwich  hour-angle  =  Greenwich  sidereal  time  —  right  ascen- 
sion of  star, 

.  f  T  =  Greenwich  hour-angle  +  D.  R.  longitude,  if  east, 
\  T  =  Greenwich  hour-angle  —  D.  R.  longitude,  if  west. 

As  an  application  of  the  Sumner  method  to  a  star  observa- 
tion, let  us  take  the  observation  of  Sirius,  Dec.  17,  1917, 
off  Cape  Agulhas,  already  treated  as  a  time-sight  (p.  105). 

From  the  preliminary  calculations  there  given,  we  have : 

Greenwich  hour-angle,  line  (11),  page  105 2h  Gm  34«  (1) 

D.  R.  longitude  (p.  105)  is  20°  41'  E.,  or  by 

Table  9  (p.  249) 1  22  44  E.  (2) 

By  formula  (8)  above,  we  add  (1)  and  (2), 

giving  T 3     29     18        (3) 

The  star  bore  west 1  (p.  105)  so  we  choose  formula  (2) 
(p.  113),  and  write : 
cos    lat.    (p.    106,    line    19),   35°  20'    S.    by    D.  R. 

(see  Table  4,  p.  231) 9.91158  (4) 

cos  dec.  (p.  106,  line  12),   -  16°  36'  (Tab.  4,  p.  212)  9.98151  (5) 

hav.  T,  3h  29m  18s  (line  3,  above)  (see  Table  10,  p.  263)  9.28872  (6) 

Adding  (4)  to  (6)  gives,  by  formula  (2),  page  1 13,  hav.Z,  9.18181 2  (7) 

1  See  p.  116,  footnote. 

2  Sum  diminished  by  20  (see  footnote,  p.  102). 


NEWER  NAVIGATION  METHODS  121 

Next  we  choose  formula  (3),  page  113,  since  latitude  and 
declination  are  both  — .     We  have : 

By  formula  (3),  lat.  -  dec.  =  35°  20'  -  16°  36'  =         18°  44'      (8) 

We  now  use  formula  (5),  page  113.     We  have: 

No.  hav.  18°  44'  (8)  (see  Table  10,  p.  254) 0.02649     (9) 

No.  hav.  X1  (7)  (see  Table  10,  p.  261) 0.15194  (10) 

Adding  (9)  and  (10)  gives  No.  hav.  Z 0.17843  (11) 

And  Z,  corresponding  to  (11)  is  found  from 

Table  10,  page  262 49°  59'    (12) 

Then,  by  formula  (6),  page  113, 

computed  alt.  =  90°  -  Z  (12),  or 40°    1'    (13) 

This  computed  altitude  (13)  must  be  compared 
with  the  observed  altitude,  fully  corrected. 

This  was  (p.  106,  line  18) 40°    2'    (14) 

Difference  between  (13)  and  (14),  in  minutes,  or  dis- 
tance of  Sumner  point  from  D.  R.  point  in  miles 
(p.  113)  1  mile  (15) 

Next  we  find  the  star's  azimuth  from  Table  11,  page  287. 

The  top  argument  for  entering  the  table  is  T,  line  (3), 
and  it  must  be  found  in  the  " afternoon"  lines,  since  the  star 
bore  W.  The  argument  for  the  left-hand  column  is  the 
declination,  line  (5).  Under  T  (p.  287),  and  opposite 
declination,  we  find  (approximately)  the  tabular  index  num- 
ber 7550.  Then  we  find  the  computed  altitude,  40°  (13), 
in  the  right-hand  column  of  the  table  (p.  289),  and  follow 
along  its  horizontal  line  until  we  again  reach  the  index 
number  7550.  The  nearest  to  7550  is  7544;  and  under 
this  number,  at  the  foot  of  the  column,  we  find  the  two 
" afternoon"  azimuths  260°  and  280°. 

These  two  numbers  are  so  nearly  equal  that  there  is  un- 
certainty in  choosing  between  them.  Had  the  observer 
taken  the  star's  bearing  by  compass  at  the  time  of  observa- 
tion (p.  115),  the  uncertainty  would  be  removed.  But 
in  the  absence  of  this  information,  we  must  have  recourse 
to  Table  12  (p.  290),  the  auxiliary  azimuth  table.  Enter- 
ing this  table  with  the  pair  of  arguments  of  the  present 

1  No.  hav.  here  obtained  from  hav.  without  finding  the  angle  X 
(p.  117,  footnote). 


122  NAVIGATION 

problem:  viz.  latitude  35°,  declination  17°,  we  find  the 
auxiliary  angle  31°.  The  computed  altitude  (13)  being 
40°,  is  greater  than  the  auxiliary  angle,  and  the  latitude  is  — . 
Therefore,  by  the  instructions  (p.  115),  the  azimuth  is 
not  between  90°  and  270°.  We  therefore  choose  280°  as 
our  final  azimuth,  since  260°,  the  other  possible  value,  is  in 
the  prohibited  area  between  90°  and  270°. 

The  computed  altitude  (13)  being  less  than  the  observed 
altitude,  this  observation  places  the  Sumner  point  1  mile 
(15)  from  the  D.  R.  point,  and  bearing  from  it  280°,  the  same 
as  the  sun's  azimuth  (p.  113).  The  traverse  table  (p.  156) 
gives,  for  distance  1  and  course  280°,  latitude  0.2,  departure 
1.0.  The  longitude  difference,  by  Table  2  (p.  172),  is  1'.2, 
for  the  departure  1 .0.  Therefore,  since  azimuth  280°  indicates 
on  the  compass  card  that  the  Sumner  point  is  W.  and  N. 
of  the  D.  R.  point,  we  have : 

lat.  of  Sumner  point  =  -  35°  20'  (4)  +  0'.2  =  -  35°  20'  (16) 
long,  of  Sumner  point  =  20°  41'  E.  (2)  -  1'.2  =  20°  40'  E.  (17) 

The  bearing  of  the  Sumner  line  will  be  90°  greater  than 
the  star's  azimuth  (p.  Ill) ;  so  we  have  : 

Bearing  of  Sumner  line  =  280°  +  90°  =  370° ;  or, 

dropping  360°  =    10°  (18) 

The  foregoing  calculation  of  the  Sumner  point  from  a 
star  observation  can  of  course  also  be  put  in  condensed  form. 
In  doing  so,  we  have  repeated  certain  numbers  from  page  107 
without  references  in  parentheses.  But  numbers  taken 
from  the  extended  calculation  just  given  have  their  reference 
numbers  attached. 

This  condensed  form,  like  the  others  previously  given,  is 
the  form  of  calculation  which  would  be  used  in  actual 
navigation.  It  is  most  important,  in  the  interest  of  numeri- 
cal accuracy,  to  make  all  calculations  upon  forms ;  and  no 
numbers  should  be  written  on  the  forms  without  having  an 
adjoining  statement  as  to  the  meaning  of  the  numbers. 


NEWER  NAVIGATION  METHODS 


123 


SUMNER   LINE,    CONDENSED   FORM.     STAR 


Watch  time : 

C.  -  W. : 
Chr.  time : 
Chr.  corr'n : 
G.  M.  T. : 

R.  A.  mean  sun : 
Corr'n,  past  noon : 
Greenw'h  sid.  time : 
R.  A.  of  Sirius : 
Greenw'h  hour-angle 

D.  R.  long. : 
T: 


16*   29"»48» 

-  1     23  50 

15      5  58 

-  2  28 

15      3  30 

17    42  10 

2  28 

8    48      8 

6    41  34 

:     2       6  34 

1     22  44  E.  (2) 

3    29  18       (3) 


Obs'd  alt. : 
Index : 
Table  6: 
Table  7 : 
Corr'd  alt. 


T  or  (24*  -  T)  1 :  3ft  29"»  18*    (3)  hav. 

Dec. :  -  16°  36'  cos : 

D.  R.  lat. :          -  35     20  cos : 

Sum  of  3  =  hav.  X  : 

No.  hav.  X : 

Lat.  -  Dec. :  18 

Sum' of  2  =  No.  hav.  Z: 

Z: 

Computed  alt.  =  90°  -  Z 

Obs'd  alt.,  corr'd: 

Diff.: 

Index  No.:  7550 

Azimuth :     280° 

Lat.  diff. :  0'.2        Dep. :  1.0       Long.  diff. :  1'.2 

Sumner  pt.  lat. :  -  35°  20'  (16) ;  long. :  20°  40'  E. 

Bearing  of  Sumner  line :   10°  (18) 


9.28872 
9.98151 
9.91158 
9.18181 
0.15194 

44'    (8); No. hav.: 0.02649 
0.17843 
49°  59' 
40     1 
40     2 
1 


40°  3' 

+  5 

-  1 

-  5 
40    2 


(6) 

(5) 

(4) 

(7) 

(10) 

(9) 

(ID 

(12) 

(13) 

(14) 

(15) 


(17) 


We  have  now,  in  the  foregoing  examples,  illustrated  the 
manner  of  determining  a  Sumner  line  completely  by  ascer- 
taining the  latitude  and  longitude  of  one  point  on  the  line 
(the  Sumner  point),  and  the  bearing  of  the  line  itself  at  that 
point.  It  may  be  desired  to  draw  the  line  on  the  chart, 
which  will  always  interest  the  navigator  if  he  is  near  the 
coast  and  has  a  large-scale  chart.  To  draw  it,  we  merely 
locate  the  Sumner  point  on  the  chart  by  its  latitude  and  longi- 

1  See  footnote,  p.  116. 


124  NAVIGATION 

tude,  and  then  draw  the  line  through  the  point  so  that  it 
will  make  with  the  meridian  an  angle  equal  to  the  bearing 
which  has  been  computed  for  the  line.  The  Sumner  line 
should  be  extended  in  both  directions  from  the  Sumner 
point,  for  any  convenient  distance,  in  such  a  way  that  the 
point  will  be  near  the  middle  of  the  line. 

We  can  now  gain  a  better  understanding  as  to  Sumner 
navigation  by  comparing  the  results  obtained  in  one  of  the 
foregoing  examples  with  the  corresponding  calculation  of 
the  same  example  as  a  time-sight.  Thus  from  the  same  ob- 
servation (pp.  104,  119) 

As  A  TIME-SIGHT  As  A  SUMNER  OBSERVATION 

From  D.  R.  latitude  42°  20'  N. ;  From  D.  R.  latitude  42°  20'  N. ; 
D.  R.  longitude  35°  16'  W.,  we  D.  R.  longitude  35°  16'  W.,  we 
found  the  ship's  longitude  to  be  found  the  Sumner  point  to  be 
35°  24'  W.  in  latitude  42°  17' ;  longitude  35° 

19'  W. ;  and  azimuth  of  Sumner 

line,  307°. 

Starting  with  the  same  observed  altitude,  and  the  same 
D.  R.  position  of  the  ship,  we  get  quite  different  results  by 
the  two  methods  of  calculation.  The  time-sight  gives  us 
nothing  but  a  longitude;  and  it  will  be  the  correct  ship's 
longitude  only  if  the  D.  R.  latitude  was  also  correct  (p.  101). 
Therefore  the  time-sight  calculation  leaves  us  with  both 
latitude  and  longitude  still  affected  by  possible  errors  in  the 
D.  R.  latitude. 

On  the  other  hand,  the  Sumner  calculation  gives  us  both 
a  latitude  and  a  longitude,  but  neither  belongs  to  the  ship's 
position.  They  both  belong  to  the  position  of  the  Sumner 
point,  but  they  are  free  from  the  effects  of  any  D.  R.  errors. 
They  fix  the  Sumner  point  only,  but  they  fix  it  correctly. 
Furthermore,  our  knowledge  that  the  ship  is  somewhere 
on  the  Sumner  line  is  also  a  fact,  free  from  error.  So  what 
we  learn  from  the  Sumner  method  is  sure ;  what  we  get  by 
the  older  methods  is  all  really  D.  R.  information  in  some 


NEWER  NAVIGATION  METHODS  125 

degree.  The  Sumner  method  is  independent  of  D.  R.,  an 
advantage  of  which  the  value  cannot  be  estimated  too  highly. 

Furthermore,  it  can  be  shown  mathematically  (cf.  p.  Ill) 
that  a  single  observation  can  never  really  do  more  than 
determine  a  line  on  which  the  ship  must  be.  Even  a  noon- 
sight  does  no  more  than  this ;  for  in  determining  the  ship's 
latitude,  it  really  only  makes  known  a  horizontal  line  (the 
ship's  latitude  parallel)  on  the  chart.  In  other  words,  for 
a  noon-sight  the  Sumner  line  is  horizontal,  or  has  a  bearing 
of  90°.  And  it  will  always  come  out  90°,  if  a  noon-sight  is 
worked  as  a  Sumner  observation. 

But  the  principal  purpose  of  our  present  comparison  of 
the  two  methods  of  calculation  is  to  warn  the  navigator 
against  falling  into  the  error  of  imagining  the  ship  to  be  at 
the  Sumner  point.  The  observation  does  no  more  than  tell 
us  where  the  Sumner  point  is,  and  that  the  ship  is  somewhere 
on  the  line ;  so  far  as  the  observation  is  concerned,  all  points 
on  the  line  are  equally  likely  to  be  the  ship's  true  position. 
Therefore  it  is  misleading  to  call  the  Sumner  point  the  ship's 
"most  probable  position."  Were  it  so,  a  second  observation, 
made  later  in  the  day,  would  give  another  "most  probable 
position"  of  the  ship.  We  should  then  be  naturally  led  to 
take  as  the  ship's  final  location  a  point  midway  between  the 
two  "most  probables,"  ascribing  their  divergence  to  possible 
errors  of  observation.  But  the  ship's  real  position  we  already 
know  (p.  Ill)  to  be  at  the  intersection  of  the  two  Sumner 
lines  resulting  from  the  two  observations.  And  this  inter- 
secting point  may  be  many  miles  from  both  "most  proba- 
bles," and  from  the  above-mentioned  midpoint  between 
them. 

Less  than  two  observations  cannot  fix  the  ship's  position 
completely;  when  two  have  been  made,  a  correct  applica- 
tion of  the  Sumner  method  requires  that  the  intersection 
point  of  two  Sumner  lines  be  determined  by  calculation. 
But  before  explaining  the  method  of  doing  this,  we  must 
describe  an  excellent  alternative  way  of  making  Sumner 


126  NAVIGATION 

calculations  such  as  we  have  given  in  the  above  examples. 
The  results  are  the  same  results  as  before,  but  they  are 
obtained  with  less  work,  and  quite  without  logarithms,  by 
means  of  special  tables  such  as  our  Table  13  (p.  292), l  which 
we  shall  call  Kelvin's  Sumner  Line  Table. 

This  table  has  a  pair  of  arguments  (p.  11),  a  and  b,  a  ap- 
pearing at  the  heads  of  the  tabular  columns,  and  b  in  the 
left-hand  column  of  each  page.  Corresponding  to  these 
two  arguments,  the  table  gives  two  angles,  K  and  Q ;  so  that 
whenever  a  and  b  are  given  we  can  find  the  corresponding 
K  and  Q ;  or,  if  a  and  K  should  be  given,  we  can  find  the 
corresponding  b  and  Q. 

In  the  Sumner  problem  we  obtain,  by  preparatory  calcu- 
lation (cf.  pp.  119,  123),  the  following  data: 

Declination  of  sun  (or  star) ;  D.  R.  latitude ;  D.  R.  longitude ; 
T,  the  ship's  apparent  time  of  the  observation  for  the  sun,  or  the 
hour-angle  for  a  star ; 

and  we  wish  to  get  the  computed  altitude  and  the  azimuth. 

The  principle  on  which  Table  13  depends  is  that  the  D.  R. 
latitude  and  longitude  being  always  somewhat  uncertain, 
we  can,  if  we  choose,  change  them  by  reasonable  amounts 
before  beginning  our  calculations.  The  Sumner  point  will 
then  be  determined  by  its  distance  and  bearing  from  the 
changed  D.  R.  point,  instead  of  the  original  D.  R.  point. 
By  this  device  the  tabular  calculation  is  much  facilitated. 
The  use  of  the  table  is  easy  after  a  little  practice,  the  work 
being  divided  into  a  series  of  separate  operations.  In  de- 
scribing these  operations  we  have  used  small  subscript  num- 
bers, to  distinguish  the  several  arguments,  etc. ;  as,  for  in- 
stance, in  Operation  1  we  use  ai,  bi,  KI. 

1  These  tables  were  first  published  by  Lord  Kelvin  in  1876. 
More  extended  ones  were  recently  issued  by  Lieutenant  de  Aquino, 
of  the  Brazilian  Navy;  and  these  were  reprinted  by  the  Hydro- 
graphic  Office,  United  States  Navy,  in  1917.  Aquino  also  improved 
Kelvin's  method  of  using  his  table. 


NEWER  NAVIGATION  METHODS  127 

OPERATION  1,  requiring  no  interpolation.  Enter  Table  13 
with: 

Arg.  ai  =  declination,  taken  without  regard  to  +  or  —  sign,  and  cor- 
rect to  the  nearest  whole  degree  only ; 
Arg.    61  =  T,  if  T  is  between  0*  and  6* ; 

=  12*  -  T1,  if  T  is  between  6*  and  12*; 
=  T  -  12*,  if  T  is  between  12*  and  18*; 
=  24*  -  r,  if  T  is  between  18*  and  24* ; 

and  before  use  61  must  be  turned  into  degrees  with 
Table  9  (p.  249).  It  need  be  correct  to  the  nearest 
degree  only.  This  proceeding  will  make  61  always 
less  than  90°. 

Then  take  from  the  table  the  tabular  angle  KI,  also  correct 
to  the  nearest  degree  only. 

OPERATION  2,  requiring  simple  interpolation.  Enter  the 
table  a  second  time  with : 

Arg.  02  =  the  Ki,  obtained  in  Operation  1. 

Then,  under  this  a2,  run  down  the  ^-column  until  you 
find  the  declination  (taken  without  regard  to  +  or  -  sign) ; 
so  that,  in  other  words,  K2  =  declination. 

Take  from  the  table  the  angle  Q2,  which  stands  next  to 
the  declination  K2)  and  also  the  62,  which  is  in  the  left-hand 
argument  column,  in  the  same  horizontal  line  with  the 
declination  K2  in  the  ^-column.  It  will  rarely  be  possible 
to  find  the  declination  (which  must  this  time  be  exact  to 
the  nearest  minute)  in  the  K-column ;  so  that  a  simple 
interpolation  will  be  necessary  in  getting  Q2  and  62.  An 
example  of  this  interpolation  will  be  found  on  page  129 ;  and, 
as  we  shall  see,  it  is  practically  the  only  numerical  calculation 
required  in  the  whole  problem.  The  Kelvin  method  is  very 
much  shorter  than  it  looks. 

The  angle  Q2  is  used  in  choosing  the  longitude  of  the 
" changed  D.  R.  point";  the  latitude  of  that  point  will  be 
found  in  Operation  3.  To  utilize  Q2  for  a  sun  observation, 
calculate  the  Greenwich  apparent  time  (G.  A.  T.)  of  the 


128  NAVIGATION 

observation,  as  on  page  102,  line  (8),  and  turn  it  into  de- 
grees with  Table  9  (page  249).     Then  : 

(1)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  —  Q2,  if,  in  Oper- 

ation 1,  T  was  less  than  6*; 

(2)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (180°  -  Q2)  if, 

in  Operation  1,  T  was  between  6*  and  12*; 

(3)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (180°  +  Q2)  if, 

in  Operation  1,  T  was  between  12*  and  18*; 

(4)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (360°  -  Q2)  if, 

in  Operation  1,  T  was  between  18*  and  24*. 

When  the  subtractions  in  these  formulas  cannot  be  made, 
the  G.  A.  T.  may  be  increased  by  360° ;  and  when  the  west 
longitude  comes  out  greater  than  180°,  subtract  it  from  360°, 
and  call  it  east  longitude. 

In  the  case  of  a  star,  we  must  use,  in  the  above  formulas, 
the  Greenwich  hour-angle,  instead  of  the  G.  A.  T.  See 
page  105,  line  (11),  for  the  method  of  obtaining  it. 

OPERATION  3,  requiring  no  interpolation.  Enter  the  table 
a  third  time  with : 

Arg.  a3  =  KI,  again  as  obtained  in  Operation  1. 

(5)  Arg.  63  =  90°  -  (62  +  changed    D.   R.   lat.),    if  latitude    and 

declination  are  of  opposite  signs,  one    +  and 
one  —  ; 

(6)  Arg.  63  =  (62  +  changed  D.  R.  lat.)  -  90°,  if  T  was  between 

90°  and  270° ; 

(7)  Arg.  63  =  90°  -  (62  -  changed  D.   R.  lat,),  if  latitude  is  less 

than  62 ; 

(8)  Arg.  63  =  90°  +  (62  -  changed    D.    R.    lat.),    if    latitude    is 

greater  than  62. 

In  choosing  among  formulas  (5)  to  (8),  give  them  pre- 
cedence in  order;  do  not  use  (7)  or  (8)  if  the  conditions 
stated  for  (5)  or  (6)  are  satisfied.  And  at  this  point,  use 
your  privilege  of  choosing  any  reasonable  changed  D.  R.  lati- 
tude for  the  ship ;  and  choose  one  that  differs  as  little  as  pos- 
sible from  the  original  D.  R.  latitude,  and  that  yet  makes 
63  a  whole  number  of  degrees.  In  this  way,  all  further 


NEWER  NAVIGATION  METHODS  129 

interpolation  is  avoided.  Having  once  chosen  among  the 
formulas,  the  latitude  is  used  without  regard  to  +  or  — 
signs. 

To  complete  Operation  3,  having  entered  the  table  with 
the  pair  of  arguments  a3  and  63,  take  out  the  tabular  K3 
and  Q3. 

Kz  is  now  the  computed  altitude,  to  be  used  (p.  113)  in 
locating  the  Sumner  point  from  the  changed  D.  R.  point; 
and  Q3  is  the  sun's  true  azimuth,  which  will  always  come 
from  the  table  less  than  90°.  If  the  ship  is  in  the  northern 
hemisphere,  this  azimuth  must  be  counted  from  the  north 
point  of  the  horizon  if,  in  Operation  3,  we  used  formulas  (6) 
or  (7) ;  or  from  the  south  point  of  the  horizon,  if  we  used 
formulas  (5)  or  (8).  With  the  ship  in  the  southern  hemi- 
sphere, interchange  the  north  and  south  points  of  the  horizon 
in  these  directions.  And  in  both  hemispheres,  the  azimuth 
will  of  course  be  counted  toward  the  east  or  west,  according 
as  the  observation  was  a  "forenoon"  or  " afternoon"  one 
(cf.  p.  120). 

We  shall  now  use  Table  13  for  the  example  given  on  page 
119  in  condensed  form.  We  have  (p.  127) : 

OPERATION  1. 

01  =  dec.  =  23°,  p.  119,  line  (1),  to  the  nearest  degree; 

&!  ==:  T  =  2h  41"1  31',  p.  119,  line  (4)  =  40°,  to  the  nearest 
degree ;  and,  with  a\  and  61  as  arguments,  Table  13  gives 
(p.  298)  :  Kj.  =  36°,  to  the  nearest  degree. 

OPERATION  2. 

02  =  Ki  =  36°. 
K2  =  23°  24',  p.  119,  line  (1) 

and,  with  02  and  K2,  we  must  find  Q2  and  62.  Running  down 
the  column  headed  a  =  36°  (p.  302),  we  find : 

When  K2  =  23°    5',  Q2   =  39°  43',  b2  =  29°, 
When  K2  =  23°  51',  Q2  =  40°    0',  b2  =  30°. 

We  wish  to  interpolate  for  7C2  =  23°  24',  which  is  19' 
down  from  23°  5'  toward  23°  51'.  The  whole  distance  from 


130  NAVIGATION 

23°  5'  to  23°  51'  is  46'.  Therefore  we  must  interpolate 
down  £f  of  the  whole  interval  from  Q2  =  39°  43'  to  Q2  = 
40°  0'.  The  difference  between  these  two  Q2's  is  17' ;  there- 
fore the  final  Q2,  belonging  to  K2  =  23°  24',  is  39°  43'  + 
if  X  17'  =  39°  43'  +  7'  =  39°  50'.  Similarly,  the  difference 
between  the  two  fr2's  being  60',  the  final  value  of  62,  for 
K2  =  23°  24',  is  29°  +  if  X  60'  =  29°  25'.  These  two 
little  interpolations  are  practically  all  the  calculation  required 
in  the  whole  problem. 

To  find  the  longitude  of  the  changed  D.  R.  point  from  the 
above  Q2  =  39°  50',  we  take  from  page  102,  line  (8), 

Greenwich  apparent  time  of  observation,  5*  2m  35» 

which,  by  Table  9  (p.  249)  is,  75°  39' 

We  now  use  formula  (1),  page  128,  because  T,  in  Opera- 
tion 1,  was  less  than  6*.     We  get : 

W.  long,  of  ch'd  D.  R.  pt.  =  G.  A.  T.  -  Q2  =  75°  39'  -  39°  50' 
=  35°  49'  W. 

OPERATION  3. 

a,  =  Ki  =  36°. 

The  D.  R.  latitude  is  +  42°  20'  (p.  119,  line  (9)) ;  and  as 
the  declination  is  — ,  we  choose  formula  (5),  page  128. 
This,  without  changing  the  D.  R.  latitude,  would  give  63  = 
90°_(&2+D.  R.lat.)  =90° -(29° 25'+ 42° 20')  =  90°-71°45'; 
but  by  choosing  a  changed  D.  R.  latitude  of  42°  35',  we  shall 
make  63  a  whole  number  of  degrees.  So  we  have: 
63  =  90°  -  (62+  changed  D.  R.  latitude)  =  90°  -  (29°  25' 
+  42°  35')  =  90°  -  72°  =  18°. 

Now  we  enter  the  table  with  the  arguments  a3  =  36°,  and 
63  =  18°,  and  obtain,  without  interpolation  (p.  302) : 

K3  =  computed  altitude  =  14°  29', 
Q3  =  sun's  true  azimuth  =  37°  22'. 

This  azimuth  must  be  counted  from  the  south  point  of 
the  horizon,  since  we  used  formula  (5)  in  Operation  3 ;  and 


NEWER  NAVIGATION  METHODS  131 

as  the  observation  was  an  afternoon  one,  the  correct  azi- 
muth will  be  S.  37°  22'  W.  (cf.  p.  19).  Counted  in  the  United 
States  Navy  way,  from  the  north  toward  the  east,  and  so 
around  to  360°,  the  azimuth  will  be  217°  22'. 

On  page  119,  we  found  :  Computed  altitude,  14°  26' ;  azi- 
muth, 217°. 

This  computed  altitude  differs  by  3'  from  the  value  just 
found  by  Table  13.  The  difference  is  due  to  our  having 
changed  the  D.  R.  point. 

From  the  changed  D.  R.  point,  in  latitude  42°  35'  N. ; 
longitude  35°  49'  W.,  we  now  calculate  (see  Condensed  Form, 
next  page)  the  position  of  the  Sumner  point  to  be :  latitude 
42°  34'  N. ;  longitude  35°  50'  W.  The  former  position,  as 
obtained  on  page  119,  was :  latitude  42°  17'  N. ;  longitude 
35°  19'  W. 

These  two  Sumner  point  positions  should  lie  on  the 
same  Sumner  line  if  the  method  of  Table  13  gives .  correct 
results ;  and  they  will  satisfy  this  test,  if  the  bearing 
of  a  line  joining  them  agrees  with  the  azimuth  of  the 
Sumner  line,  which  is  217°  +  90°  =  307°.  From  the  two 
Sumner  point  positions  we  have :  latitude  difference  =  17' ; 
longitude  difference  =  31';  departure  (Table  2,  p.  174) 
=  23.0.  The  traverse  table  (p.  164)  gives,  for  latitude  17, 
departure  23.0,  the  distance  28,  course  307°.  The  agree- 
ment is  perfect,  and  shows  that  the  same  Sumner  line 
passes  through  both  points,  though  they  are  28  miles 
apart.  This  test  also  shows  that  the  calculation  may 
indicate  any  point  on  the  Sumner  line  as  the  Sumner  point, 
if  the  D.  R.  position  of  the  ship  is  uncertain :  and  so 
we  again  call  attention  to  the  error  of  taking  the  cal- 
culated Sumner  point  as  the  ship's  most  probable  position 
(cf.  p.  125). 

We  now,  as  usual,  repeat  the  above  calculation  by  Table  13, 
in  condensed  form,  and  including  the  final  determination 
of  the  position  of  the  Sumner  point  from  the  changed  D.  R. 
point. 


132  NAVIGATION 

SUMNER  LINE  BY  TABLE  13,  CONDENSED  FORM.    SUN 

[The  following  is  taken  from  page  119.] 


Decl.,  4*: 

-  23°  23'.7 

Eq 

.  of  time  :  +  3m 

22».3 

H.  D.  : 

0.1 

H. 

D.  : 

1.2 

Decl.,  4*59«: 

-23     24 

Eq 

.  time  :       +  3 

21.1 

Watch  time  : 

2h   29"» 

58* 

Obs'd  alt.  : 

14°   19' 

C.   -  W.  : 

2     27 

8 

Index  : 

+  4 

Chr.  time  : 

4     57 

6 

Table  6  : 

+  12 

Chr.  corr'n  : 

+  2 

8 

Table  7  : 

-5 

G.  M.  T.: 

4     59 

14 

Corr'd  alt.  : 

14     30 

Eq.  of  time  : 

+  3 

21 

D.  R.  lat.  : 

42°   20'  N. 

G.  app.  time  : 

5       2 

35 

D.  R.  long.  ; 

:  35°   16'  W. 

D.  R.  long.  : 

2     21 

4  W. 

(3) 

Ship's  app.  time, 

T:     2     41 

31 

(4) 

[The  following  is  calculated  with  Table  13.] 

OPERATION  1  OPERATION  2 

at   =  dec.    =23°  a*  =     Ki  =  36° 

61  =  T       =2*  41"*  31«(4)  K2  =  dec.   =  23°  24' 

=  40°  Table  13,  Q2  =  39°  50' 

Table  13,  Ki  =  36°  Table  13,  62  =  29°  25' 

Greenwich  app.  time   =  5&  2™  35«  =  75°  39' 
By  page  128,  form.  (1),  W.  long,  of  changed  D.  R.  pt.  =  G.  A.  T.  -  Qt 

=  35°  49'  W. 
Lat.  of  changed  D.  R.  pt.  =  42°  35'  N. 

OPERATION  3 
as  =  Ki  =  36° 

&3  =  90°  -  (62  +  changed  D.  R.  lat.)    =  18° 
Table  13,  K3  =  comp'd  alt.  =  14°    29' 

Table  13,  Qa  =  azimuth  of  sun  =  37°    22' 

or,  by  U.  S.  Navy  =  217°  22' 

Azimuth  of  Sumner  line  =  217°  22'  +  90° 

=  307°  22' 

Dist.  of  Sumner  pt.  from  changed 

D.  R.  pt.  =  corr'd  obs'd  alt.   —  comp'd  alt.          =    1'  or  1  mile 
Bearing  of  Sumner  pt.  from  changed  D.  R.  pt.  =217°, 
since  comp'd  alt.  is  less  than  obs'd  alt. 

Dist.  1,  on  course  217°,  gives  lat.  diff.,  0'.8  ;   dep.,  0.6  ;  long,  diff.,  0'.8 
Lat.  of  Sumner  pt.       =  lat.  of  ch'd  D.  R.  pt.  -  lat.  diff.      =  42°  34'  N. 
Long,  of  Sumner  pt.     =  long,  of  ch'd  D.  R.  pt.  +  long.  diff.  =  35°  50'  W. 

A  practised  navigator  can  make  the  above  complete  calcu- 
lation in  a  few  minutes,  as  there  are  no  logs  used  ;  and  any 
one  can  easily  obtain  the  necessary  practice  at  sea  by  simply 
forming  the  habit  of  working  his  sights  both  as  time-sights 
and  as  Sumners.  To  illustrate  the  subject  further,  we  now 
give,  in  condensed  form,  the  Star  Example  of  p.  123,  worked 
by  Table  13. 


NEWER  NAVIGATION  METHODS  133 

SUMNER  LINE  BY  TABLE  13,  CONDENSED  FORM.    STAR 

[The  following  is  taken  from  page  123.] 

Watch  time:  16*  29™  48«  Obs'dalt.:  40°     3' 

C.  -  W. :  -  1     23     50  Index :  +  5 
Chr.  time :  15       5     58  Table  6 :  -  1 
Chr.  corr'n  :  -  2     28  Table  7  :  -  5 
G.  M.  T. :  15       3     30  Corr'd  obs'd  alt. :  40       2 
R.  A.  mean  sun :  17     42     10 

Corr'n,  past  noon :  2  28  Dec.  of  Sirius :            - 16     36 

Greenwich  sid.  time :  8  48  8                     D.  R.  lat. :                   -35     20 

R.  A.  of  Sirius :  6  41  34 

Green,  hour-angle :  2  6  34 

D.  R.  long. :  1  22  44  E. 
T:  3  29  18 

[The  following  is  calculated  with  Table  13.] 

OPERATION  1  OPERATION  2 

ai  =  dec.    =17°  02  =    Ki   =  49° 

6,  =  T       =    3*   29"*  18«  K2  =  dec.   =  16°  36' 

=  52°  Table  13,  Q2    =51°   57' 

Table  13,  Ki   =  49°  Table  13,  b     =  25°  49' 

By  page  128,  form.  (1), 

W.  long,  of  changed  D.  R.  pt.   =  Green,  hour-angle  —  Qzl 

339°  41' 
20°     19' E. 
Lat.  of  changed  D.  R.  pt.  =   -  35°     49' 

OPERATION  3 
a,  =  Ki  =  49° 

By  form.  (8),  page  128,  bs     =  90°  +  (61  -  changed  D.  R.  lat.)   =  80° 
Table  13,  Kt  =  comp'd  alt.  =          40°  15' 

Table  13,   Q»  =  az.  of  Sirius  =  N.    81°  25'  W. 

or,  by  U.  S.  Navy  =        278°  35' 
Az.  of  Sumner  line  =        368°  35',  or  8°  35' 

Dist.  of  Sumner  pt.  from  changed 

D.  R.  pt.  =  corr'd  obs'd  alt.  —  comp'd  alt.  =  —  13'  or  13  miles 

Bearing  of  Sumner  pt.  from  changed  D.  R.  pt.      =  99°, 

since  comp'd  alt.  is  greater  than  obs'd  alt. 

Dist.  13,  on  course  99°,  gives  lat.  diff.,  2'.0 ;   dep.,  12.8 ;  long,  diff .,  15'.9 

Lat.  of  Sumner  pt.     =  lat.  of  ch'd  D.  R.  pt.  -f  lat.  diff.          35°  51' 

Long,  of  Sumner  pt.  =  long,  of  ch'd  D.  R.  pt.  +  long.  diff.   -      20°  35'  E. 

To  complete  this  part  of  our  subject,  it  remains  to  show 
how  the  position  of  the  ship  can  be  found  at  the  intersec- 
tion of  two  Sumner  lines  (pp.  Ill,  125)  resulting  from 
two  different  observations.  Figure  18  explains  the  nature  of 
the  problem;  and  it  is  almost  exactly  the  same  figure  and 

1  #2  being  larger  than  the  Greenwich  hour-angle,  the  latter  was 
increased  by  360°,  to  make  the  subtraction  possible  (p.  128). 


134 


NAVIGATION 


problem  treated  in  Chapter  V,  when  we  discussed  fixing  a 

ship's   position   by   means   of    " bearings   from   the   bow" 

(p.  54). 
The  two  Sumner  lines  in  Fig.  18  are  SL  and  S'L,  passing 

through  the  two  Sumner  points  S  and  Sf,  whose  latitudes 

and  longitudes  are  known 
by  calculation  from  the 
observed  altitudes.  The 
bearings  or  azimuths  of  the 
two  Sumner  lines  from  the 
north  are  the  two  angles 
NSL  and  N'S'L,  which  are 
also  known  from  the  pre- 
vious calculations.  It  is 
now  required  to  find  the 
latitude  and  longitude  of 
the  intersection  point  L, 
where  the  ship  is  situated. 
The  similarity  of  this 
problem  to  the  former  one 
/^  in  Chapter  V  becomes  plain, 

F,o.  lS.-Interaection  of  Sumner  Lines.  if  we  imaSine  a  second  shiP 

sailing  from   one   Sumner 

point  to  the  other,  as  from  S  to  $',  and  taking  bearings 
from  her  bow  upon  our  ship,  located  at  L.  These  bearings 
will  be  the  two  angles  S'SL  and  S"S'L.  If  the  second 
of  these  angles  should  happen  to  be  just  twice  as  big 
as  the  first,  the  distance  S'L  between  the  two  ships  at 
the  time  of  the  second  bearing  would  be  equal  (p.  54)  to 
the  distance  SSf  run  by  the  imagined  ship  between  the  two 
observations. 

This  would  enable  us  to  fix  the  position  of  the  imagined 
ship  at  £',  if  L  were  a  lighthouse  ashore.  But  if  L  is  our 
ship,  and  S'  a  Sumner  point  of  known  position,  the  same 
observations  of  bow  bearings  would  fix  the  position  of  our 
ship  at  L.  Nor  is  it  necessary  (or  possible)  to  measure 


NEWER  NAVIGATION  METHODS  135 

such  imaginary  bearings,  or  read  the  patent  log  to  get  the 
distance  run  by  an  imagined  ship. 

For  the  distance  and  bearing  of  the  second  Sumner  point 
from  the  first  can  be  obtained  from  their  known  latitudes 
and  longitudes  with  the  traverse  table.  Thus  the  line  SS' 
(marked  " distance")  and  the  bearing  (or  course)  angle 
NSS'  become  known.  Furthermore,  the  "bow  bearing"  at 
S  is  the  angle  S'SL,  and  it  is  equal  to  the  difference  NSL  — 
NSS'.  We  have  just  seen  that  NSS'  is  obtained  from  the 
traverse  table ;  and  NSL  is  the  calculated  azimuth  of  the 
Sumner  line  through  S.  In  a  similar  way  we  get  the  other 
"bow  bearing"  SffS'L.  If  this  were  twice  the  first  one,  the 
"required  distance"  S'L  in  the  figure  would  be  equal  to  the 
known  distance  SS'  between  the  two  Sumner  points.  If 
not,  it  can  be  easily  shown  mathematically  that : 

(1)  Required  distance  =  known  distance  X  a  factor, 

(2)  log  factor  =  sin  S'SL  -  sin  (S"S'L  -  S'SL). 

By  these  simple  formulas  the  required  distance  S'L  might 
be  found :  and  as  we  also  know  the  latitude  and  longitude 
of  the  Sumner  point  S',  and  the  azimuth  or  bearing  of  S'L, 
the  traverse  table  will  make  known  the  latitude  and  longi- 
tude of  the  ship  at  L.  It  is  to  be  noted  also  that  as  we  are 
at  liberty  to  call  either  of  the  Sumner  points  S',  it  is  desirable 
to  call  that  one  S'  which  has  the  larger  "bow  bearing," 
so  that  there  will  be  no  difficulty  about  subtracting  S'SL 
from  S"S'L. 

The  factor  of  formula  (2)  above  can  practically  always 
be  found  in  our  Table  14,  the  Sumner  Intersection  Table, 
without  using  logarithms.  The  pair  of  arguments  of  the 
table  are  the  smaller  "bow  bearing"  and  the  larger  "bow 
bearing";  the  tabular  number  is  the  factor  of  formula  (1) 
above,  and  will  always  give  the  distance  of  the  intersection 
point  from  that  one  of  the  two  Sumner  points  for  which 
the  bow  bearing  was  the  larger. 

And  it  should  not  be  forgotten  that  the  Sumner  line  really 


136  NAVIGATION 

extends  equally  in  both  directions  (p.  124)  from  the  Sumner 
point,  whereas,  in  Fig.  18,  we  have  extended  it  mainly 
in  the  direction  of  the  intersection  point  L.  Now  the  cal- 
culated azimuth  of  any  Sumner  line  may  be  changed  180° 
at  will,  because  the  bearings  of  the  two  ends  of  the  line  from 
the  Sumner  point  differ  by  180°,  and  we  may  take  the  bear- 
ing of  the  line  to  be  the  bearing  of  either  end  from  the  Sumner 
point  in  the  middle  of  the  line.  Figure  18  shows,  however, 
that  for  the  purpose  of  the  present  problem  we  must  choose 
the  bearing  of  that  end  of  the  line  which  is  nearest  the  point 
of  intersection  L ;  nor  does  the  choice  ever  offer  difficulty, 
because  the  known  D.  R.  position  of  the  ship  at  L,  when 
compared  with  the  known  positions  of  the  two  Sumner 
points,  will  always  indicate  whether  L  bears  east  or  west 
of  either  Sumner  point,  and  also  whether  it  bears  north  or 
south.  And  the  bearing  of  L  once  chosen,  we  can  always 
find  either  of  the  two  bow  bearings  by  this  formula : 

(3)  Bow  bearing  =  bearing  of  Sumner  line  minus  bearing 
of  the  second  Sumner  point  Sf  from  the  first  point  S. 

In  using  formula  (3)  it  is  allowable  to  increase  the  bear- 
ings of  the  Sumner  lines  by  360°,  when  necessary  to  make 
the  subtractions  possible,  and  if  the  formula  brings  out  bow 
bearings  larger  than  180°,  subtract  them  from  360°,  and 
proceed  as  before. 

It  is  also  always  desirable  to  draw  a  rough  sketch  for 
every  intersection  problem  occurring  on  shipboard  so  as  to 
guard  against  accidental  large  errors  like  90°  or  180°  in  ob- 
taining the  two  bow  bearings ;  and  also  to  make  sure  that 
the  latitude  and  longitude  of  the  intersection  point  L  are 
correctly  computed  with  the  traverse  table. 

The  foregoing  assumes  that  the  ship  did  not  move  from 
the  point  L  between  the  two  sextant  observations  from  which 
the  two  Sumner  lines  were  calculated.  This  will  rarely 
be  the  case,  because  it  is  very  desirable  that  the  two  observa- 
tions, if  they  are  both  sun  observations,  be  separated  by 


NEWER  NAVIGATION  METHODS  137 

three  or  four  hours,  if  possible.  The  condition  of  an  unmov- 
ing  ship  will  occur  only  if  she  is  a  sailing  vessel  becalmed, 
or  a  steamer  at  anchor ;  or  if  the  two  observations  are  made 
at  nearly  the  same  time  upon  two  different  heavenly  bodies, 
such  as  two  stars. 

High  accuracy  in  the  resulting  "fix"  (p.  53)  of  the  ship 
will  then  be  attained,  if  the  azimuths  of  the  two  stars  differ 
by  about  90°  at  the  time  of  observation.  The  same  favor- 
able condition  will  be  secured  if  one  of  the  observations  is 
made  upon  a  star  near  upper  transit  (pp.  89,  96),  in  the 
twilight  just  before  sunrise  or  after  sunset;  and  the  other 
observation,  at  nearly  the  same  time,  upon  the  sun,  when 
it  is  about  12°  or  15°  above  the  horizon. 

But  if  the  ship  has  traveled  a  considerable  distance  between 
the  two  observations,  it  is  necessary  to  allow  for  such  travel 
before  calculating  the  intersection  point.  Suppose  she  has 
gone  a  distance  D,  upon  a  course  (7,  by  D.  R.,  between  the 
two  observations.  Then  simply  find  from  Tables  1  and  2 
the  difference  of  latitude  and  longitude  corresponding  to 
distance  D  and  course  C ;  and  apply  them  as  corrections  to 
the  latitude  and  longitude  of  the  Sumner  point  belonging 
to  the  first  observation.  Everything  else,  including  the 
bearing  of  the  first  Sumner  line,  remaining  unchanged, 
the  calculation  then  proceeds  by  Table  14,  just  as  if  the 
ship  had  not  moved.  The  computed  intersection  point  is 
then  the  ship's  position  at  the  time  of  the  second  sextant 
observation. 

We  shall  now  work  some  intersection  examples. 

Suppose  we  have  two  Sumner  lines,  as  shown  in  the  rough 
sketch,  Fig.  19,  taken  on  board  a  ship  becalmed.  The 
two  sextant  observations  give : 

FOR  ONE  SUMNER  POINT,  S  FOR  THE  OTHER  POINT,  Sf 

lat.»:     42°34/N.  42°     50' N. 

long.:     35°50/W.  35°     36' W. 

bearing  of  Sumner  line :  307°  93°  (changed  to  273°) 

1  As  found  on  page  132. 


138 


NAVIGATION 


The  rough  sketch,  Fig.  19,  having  been  made,  and  the 
two  "bow  bearings"  marked  with  little  circular  arcs  as 
shown,  we  call  that  one  of  the  two  Sumner  points  S't  which 
has  the  larger  bow  bearing ;  and,  for  the  point  Sf,  we  change 


FIG.  19.  —  Bough  Sketch  of  Sumner  Intersection. 

the  bearing  of  the  Sumner  line  from  93°  to  180°  +  93°  = 
273°,  so  as  to  count  the  bearing  for  that  end  of  the  line  which 
is  toward  the  intersection  point  L  (p.  136).  The  other 
bearing,  307°,  for  the  point  S,  is  already  correctly  counted. 

We  now  have,  from  the  two  Sumner  point  latitudes  and 
longitudes  :  latitude  difference  =  16' ;  longitude  difference  = 
14' ;  departure  (Table  2,  p.  174,  for  middle  latitude  43°)  = 
10.2 ;  and,  for  latitude  difference  =  16,  departure  =  10.2, 
we  find  (Table  1,  p.  162),  distance  =  19,  course  =  32°.  The 
distance  between  the  two  Sumner  points  is  therefore  19 
miles,  and  the  bearing  of  Sf  from  S  is  32°. 

Now  we  apply  formula  (3),  page  136,  and  find : 

Smaller  bow  bearing  at  S  =  307°  -  32°  =  275°. 
Larger  bow  bearing  at  S'  =  273°  -  32°  =  241°. 

Being  larger  than  180°,  these  must  be  subtracted  from 
360°  (p.  136),  giving: 

Smaller  bow  bearing  =  85°;    Larger  bow  bearing  =  119°. 

Next  we  refer  to  Table  14,  and  find  with  the  smaller 
bearing  85°,  and  the  larger  119°  the  factor  1.78  (p.  322). 


NEWER  NAVIGATION  METHODS  139 

According  to  formula  (1),  page  135,  we  then  have: 
Required  distance  LS'  =  distance  SS'  X  factor 
=  19X1.78  =  33.8  miles. 

Therefore  the  position  of  the  ship  at  L  is  distant  33.8 
miles  from  S',  and  she  bears  273°.  With  this  distance  and 
bearing  or  course  angle,  the  traverse  table  (p.  154)  gives : 
latitude  =  1.8,  departure  =  33.8.  For  the  departure  33.8, 
Table  2  gives,  for  the  middle  latitude  43°  (p.  174),  differ- 
ence longitude  =  46'.2.  The  bearing  273°  showing  that  the 
intersection  point  L  is  N.  and  W.  of  Sf,  we  have : 

Latitude     of  ship  at  L  =  42°  50'  N.  +    1'.8  =  42°  51'.8  N. 
Longitude  of  ship  at  L  =  35°  36'  W.  +  46/.2  =  36°  22'  W. 

As  a  second  example  take  the  following  two  Sumner  lines, 
as  shown  in  the  rough  sketch,  Fig.  20.  The  two  sextant 
observations  give : 

FOR  ONE  SUMNER  POINT,  S  FOR  THE  OTHER  POINT,  S' 

lat. :   14°  26'  N.  15°  30'     N. 

long. :  77°     8'  W.  76°  22'.5  W. 

bearing  of  line  :  53°  135° 

And  suppose  the  ship,  in  the  interval 
between  the  two  sextant  observations,  has 
traveled  a  distance  D  =  31  miles,  on  course 
C  =  205°.  We  must  begin  (p.  137)  by 
shifting  the  first  Sumner  point  S  a  dis- 
tance D,  on  the  course  C.  For  this  course 
and  distance,  we  have  (Table  1,  p.  160) : 
lat.,  28M;  dep.,  13.1;  diff.  long,  13'.5  FIG.  2o.-Rough 

(Table  2,  p.  168).  Sketch  of  Sumner 

Therefore,  the  latitude  and  longitude  of 
the  first  Sumner  point  must  be  corrected  (p.  137)  as  follows  : 

For  the  point  S,  lat.  =  14°  26'  N.  -  28M  =  13°  58'  N. 

long.  =  77°    8'  W.  +  13'. 5  =  IT  21'.5  W. 
Bearing  (unchanged)   =  53°. 

We  now  have,  for  the  two  Sumner  points :  lat.  diff,  92' ; 


140  NAVIGATION 

long,  diff.,  59' ;  dep.,  57.0  (p.  169) ;  dist.,  108  miles  (p.  162) ; 
bearing  of  S'  from  S,  32°. 

Now  we  have,  by  formula  (3),  page  136 : 

Smaller  bow  bearing  at  S  =    53°  -  32°  =   21°. 
Larger  bow  bearing  at  S'  =  135°  -  32°  =  103°. 

Table  14  (p.  319)  gives  the  factor  0.36 ;  so  that  the  ship  at 
L  is  distant  from  S'  108  X  .36  =  38.9  miles,  and  bears  135°. 
For  this  distance  and  bearing  we  have  (Table  1,  p.  166), 
latitude  =  27'. 6;  departure  =  27.6;  and  longitude  differ- 
ence (Table  2,  p.  168)  =  28'.6.  Finally,  then,  at  the  time 
of  the  second  sextant  observation,  the  ship  at  L  was  in 
latitude  15°  30'  N.  -  27'.6  =  15°  2'.4  N. ;  and  in  longitude 
76°  22'.5  W.  -  28'.6  =  75°  54'  W. 


CHAPTER  X 
A  NAVIGATOR'S  DAY  AT  SEA 

THE  present  chapter  contains  a  number  of  examples  by 
means  of  which  the  reader  can  gain  facility  in  the  use  of  the 
methods  set  forth  in  the  preceding  pages. 

The  steam  yacht  Nav  is  bound  from  New  York  to 
Colon,  and  the  captain  plans  to  take  his  departure  from 
the  Sandy  Hook  Lightship,  on  Dec.  18,  1917,  as  early  as 
possible  in  the  morning. 

The  first  bit  of  navigation,  to  be  accomplished  before  the 
yacht  leaves  her  anchorage  in  the  "Horseshoe,"  is  to  ascer- 
tain by  D.  R.  methods  the  proper  course  to  steer  from 
Sandy  Hook.  A  glance  at  the  track  chart  of  the  north 
Atlantic  shows  that  she  must  go  by  way  of  Crooked  Island 
Passage,  and  the  Windward  Passage  between  Cuba  and 
Haiti.  It  is  also  apparent  from  the  chart  that  the  first  land 
to  be  sighted  among  the  islands  is  Watlings  Island,  and  that 
the  proper  course  should  pass  to  the  eastward  of  it. 

The  position  of  Sandy  Hook  Lightship  *  is  lat.  40°  28'  N. ; 
long.  73°  50'  W.  Hinchinbroke  Rock,  at  the  southern  end 
of  Watlings  Island,  is  in  lat.  23°  57'  N. ;  long.  74°  28'  W. 
But  the  course  should  be  shaped  for  a  point  about  12  miles 
east  of  Watlings  Island,  to  be  perfectly  safe.  The  position 
of  such  a  point  is  (approximately)  lat.  23°  57'  N. ;  long.  74° 
15'  W.2 

1  There  is  an  excellent  list  of  latitudes  and  longitudes  in  Bow- 
ditch's  "Navigator." 

2  The  difference  between  this  longitude  and  that  of  Hinchinbroke 
Rock  is  13' ;  but  13'  here  corresponds  to  about  12  miles,  on  account 
of  Table  2. 

141 


142  NAVIGATION 

ABSTRACT  OF  LOG.    Steam  Yacht  NOD,  Dec.  18,  1917 


PATENT 
LOG 

COMPASS 
COURSE 

TRUE 
COURSE 

7  :  02  A.M. 

Took  departure  from  Sandy 

Hook  Lightship  

26.2 

S. 

188° 

7:21 

Sunrise,  observed  azimuth 

31.0 

S. 

188° 

8:00 

41.0 

S. 

188° 

9:00 

57.2 

S. 

188° 

9:36 

Bow  bearing,  Barnegat  .... 

67.0 

S. 

188° 

9:42 

Altitude  and  azimuth  

69.1 

S. 

188° 

9:57 

Beam  bearing,  Barnegat  .  .  . 

72.5 

S. 

188° 

(fix,  lat.  39°  45'  N.  ;   long. 

73°  59'  W.) 

-•• 

10:00 

73.4 

S. 

188° 

10:07 

Changed  course  

75.3 

S.|E. 

182° 

11:00 

88.7 

S.|E. 

182° 

11:42 

Ex-mer.  obs'n  lat,  39°  19'; 

D.  R.  long.  73°  58' 

98.5 

S.£E. 

182° 

12:00 

102.6 

S.|E. 

182° 

1  :  00  P.M. 

117.7 

S.^E. 

182° 

2:00 

133.0 

S.£E. 

182° 

3:00 

149.0 

S.|E. 

182° 

4:00 

163.8 

S.£E. 

182° 

4:12 

Alt.  and  az.,  fix,  lat.  38°  1  1'  ; 

long.  73°  54'  

166.9 

S.iE. 

182° 

5:00 

182.0 

*-*•  2  •*"* 

S.£E. 

182° 

6:00 

197.2 

S.fE. 

182|° 

By  the  method  of  page  20,  the  course  from  Sandy  Hook 
Lightship  should  be  181°,  and  the  distance  is  990  miles. 
These  numbers,  and  all  subsequent  numbers  in  the  present 
chapter,  should  be  verified  by  the  reader. 

The  distance  being  quite  large,  it  is  well  to  check  it  by 
the  logarithmic  method,  page  33.  The  result  by  this  method 
is:  course  181°  14',  distance  991.7  miles. 

The  chart  also  shows  that  this  course  will  carry  the  yacht 
very  near  Barnegat  Light,  on  the  coast  of  New  Jersey.  The 
position  of  this  light  is  lat.  39°  46'  N. ;  long.  74°  6'  W.  The 
captain  decides  that  it  will  be  well  to  plan  passing  this  light 


A  NAVIGATOR'S  DAY  AT   SEA  143 

at  about  5  miles'  distance.  The  position  of  a  point  5  miles 
east  of  Barnegat  Light  is  lat.  39°  46'  N.,  long.  73°  59'  W.  The 
course  and  distance  to  this  point  from  Sandy  Hook  Ship 
are  189°  and  42.5  miles.  This  course  is  so  nearly  the  same 
as  the  course  to  Watlings  Island  that  the  captain  decides 
to  steer  the  189°  course. 

All  this  work  must  be  complete  before  reaching  Sandy 
Hook,  for  the  course  from  the  lightship  must  be  ready  for 
the  quartermaster  before  the  lightship  is  passed.  And 
there  is  still  more  preliminary  work.  For  the  courses  cal- 
culated above  are  true  courses  (p.  43)  and  the  quarter- 
master must  have  the  compass  course,  so  that  he  may  be 
able  to  steer  the  yacht.  The  method  of  calculating  the 
compass  course  from  the  true  course  is  given  on  page  48 ;  and 
in  applying  it  the  captain  must  have  his  deviation  tables 
at  hand.  We  shall  assume  that  the  tables  printed  on  pages  48 
and  49  were  the  ones  furnished  by  the  compass  adjuster  for 
the  present  voyage. 

An  examination  of  the  Atlantic  track  chart  shows  that  in 
the  vicinity  of  Sandy  Hook,  the  variation,  V,  is  10°  W.,  or 
—  10°.  By  formula  (3)  (p.  49),  we  then  have,  since  the  true 
course  T  is  189°  : 

Magnetic  course  =  M  =  T  -  V  =  189°  -  (-  10°)  =  199°. 

The  second  deviation  table  (p.  49)  shows  that  when  the 
magnetic  course  (or  magnetic  bearing  of  ship's  head)  is  199°, 
the  deviation,  D,  is  +  18°.  Then,  with  V  =  -  10°,  D  =  18°, 
formula  (1),  page  45,  gives : 

Compass  error  =  E  =  V  +  D  =  -  10°  +  18°  =  +  8°. 

And  from  formula  (2),  page  45  : 

Compass  course  C=T-E=  189°  -  8°  =  181° ; 

and  so  the  yacht  must  be  steered  on  a  181°  compass  course 
for  Barnegat.  But  the  quartermaster  is  to  steer  by  "  points  " 
so  that  the  course  nearest  the  181°  course  is  due  south.  The 
captain  decides  to  have  the  yacht  steered  due  south  by 


144  NAVIGATION 

compass,  and  is  prepared  to  give  the  quartermaster  his 
orders  as  soon  as  Sandy  Hook  Lightship  shall  be  reached. 

The  foregoing  preliminary  work  having  been  completed 
the  previous  day,  the  anchor  is  tripped  at  the  Horseshoe 
about  an  hour  before  daylight  on  Dec.  18,  the  weather  being 
fine,  sea  smooth,  and  wind  light  from  the  northwest.  The 
lightship  is  reached  and  passed  at  7  :  02  A.M.,  ship's  time,  civil 
reckoning,  the  ship  then  taking  her  departure.  At  that 
moment,  the  patent  log  is  read,  and  found  to  register  26.2 
miles.  The  quartermaster  gets  his  orders  to  steer  south; 
and  all  the  above  facts  are  duly  recorded  in  the  log-book. 
And  at  every  hour  thereafter,  8,  9,  10,  etc.,  a  similar  record 
must  be  made  in  the  log-book. 

The  next  event  is  sunrise,  which  occurs  at  7 :  21,  very 
soon  after  leaving  the  lightship.  The  sun's  compass  bearing 
can  then  be  very  conveniently  observed,  and  will  furnish 
an  excellent  check  on  the  compass  adjuster.  This  observa- 
tion was  made  at  7:21  A.M.,  ship's  time,  civil  reckoning, 
corresponding  to  19*  21m,  Dec.  17,  ship's  apparent  time, 
astronomic  reckoning;  and  the  sun's  bearing  or  azimuth 
was  113°  by  compass.  This  was  entered  in  the  log-book, 
and  at  the  same  time  the  patent  log  was  read,  and  found  to 
be  31.0  miles. 

To  check  the  deviation  table,  the  procedure  was  then  as 
follows : 

By  patent  log  the  yacht  had  proceeded  from  the  light- 
ship a  distance  of  31.0  —  26.2  =  4.8  miles,  on  a  compass 
course  of  180°,  or  true  course  of  188° ;  by  D.  R.,  she  had 
therefore  reached  the  position  lat.  40°  23'  N. ;  long.  73°  51'  W. 
The  sun's  declination,  from  the  almanac,  is  —  23°  23',  and 
the  (approximate1)  T  (p.  100)  is  19*  21W.  The  sun's  true 
azimuth  is  found  from  Table  11  to  be  121°  ;  and  in  using  the 
table  for  this  purpose  take  the  altitude  of  the  sun,  for  the 

1  If  there  is  any  chance  of  this  T  being  much  in  error,  the  cap- 
tain's watch,  by  which  the  observation  is  timed,  must  be  compared 
with  the  chronometer.  See  p.  94. 


A  NAVIGATOR'S  DAY  AT   SEA  145 

moment  of  sunrise,  to  be  0°.  The  observed  compass  azi- 
muth having  been  113°,  formula  (2),  page  45,  gave  E  =  T—C 
=  121°  -  113°  =  +  8°.  Then  from  formula  (1),  page  45, 
D  =  E  -V  =  +  8  °-  (-  10°)  =  +  18°.  As  expected,  this 
deviation  agrees  with  the  deviation  table,  which  would 
not  be  likely  to  go  wrong  so  soon  after  the  beginning  of  a 
voyage. 

At  8  A.M.  the  patent  log  read  41.0;  and  at  9  A.M.,  57.2. 
The  course  was  still  S.  by  compass,  or  188°,  true  course. 

At  9  :  24  Barnegat  Light  was  sighted  by  the  lookout,  and 
the  mate  was  ordered  to  take  bow-and-beam  bearings  (p.  55) 
upon  it. 

At  9 : 36,  the  light  bore  225°  by  compass,  or  45°  from  the 
bow ;  patent  log,  67.0. 

At  9*  42m  28*  by  his  watch  the  captain  took  the  altitude 
of  the  sun's  lower  limb  with  the  sextant,  and  found  it  to 
be  18°  51'.  Index  correction  was  +  3',  and  height  of  eye, 
15  feet.  C.  —  W.  was  4A  51m  50* ;  and  the  chr.  correction 
by  the  rate  card  was  4',  slow.  Patent  log,  69.1.  At  9 : 45 
by  the  watch,  the  sun's  azimuth  was  again  observed  with 
pelorus,  and  found  to  be  137°,  compass  bearing.  It  was 
intended  to  work  a  Sumner  line  from  the  altitude  by  Kelvin's 
table;  and  the  pelorus  observation  was  made  because  the 
sun's  true  azimuth  always  comes  out  as  a  by-product,  when 
Kelvin's  table  is  used,  and  so  it  is  just  as  well  to  have  an- 
other check  on  the  deviation  table.  This  is  the  peculiar 
advantage  of  Kelvin's  table.  Without  any  additional  cal- 
culations, the  compass  is  always  checked  up  on  the  very 
course  the  ship  is  steering.  This  is  just  what  the  good 
navigator  wants. 

The  observations  could  not  be  worked  up  at  once,  be- 
cause the  captain  wished  to  see  the  result  of  the  mate's 
bow-and-beam  bearings.  At  9  :  57  by  the  watch,  Barnegat 
bore  abeam,  on  the  starboard  hand,  or  270°  by  compass,  the 
yacht  being  still  on  the  180°  compass  course.  Patent  log 
now  72.5. 


146  NAVIGATION 

Between  the  bow-and-beam  bearings  the  run  by  log  was 
72.5  —  67  =  5.5  miles.  Therefore  the  yacht  is  now  5.5 
miles  from  Barnegat  Light,  and  the  compass  bearing  of  the 
light  is  270°.  The  compass  error  being  -j-  8°,  the  true  bear- 
ing of  the  light  is  278° ;  and  the  bearing  of  the  yacht  from 
the  light  is  the  former  bearing  reversed,  or  278°  —  180°  =  98°, 
true.  From  this  comes  an  accurate  and  complete  position 
of  the  yacht.  Barnegat  Light  is  in  lat.  39°  46'  N. ;  long.  74°  6' 
W.  The  yacht,  5.5  miles  away  on  the  bearing  98°,  must,  by 
traverse  table,  be  in  lat.  39°  45'  N. ;  long.  73°  59'  W. 

At  10  A.M.,  the  log  was  73.4,  course  188°,  true. 

Now  the  captain  prepared  to  shape  a  new  course  to  be 
followed  from  the  Barnegat  bow-and-beam  bearing  "fix"  in 
the  above  lat.  39°  45'  N. ;  long.  73°  59'  W.,  at  9  :  57. 

Allowing  ten  minutes  to  work  up  the  new  course,  the 
captain  plans  to  change  course  at  10 :  07.  At  that  time 
the  ship,  on  her  course  of  188°,  will  be  (at  15-knot  speed) 
2'. 5  S.  and  practically  0'  W.  of  the  Barnegat  position.  So 
the  course  will  be  changed  when  the  yacht  is  in  lat.  39°  42'  N.  ; 
long.  73°  59'  W.,  at  10  :  07.  The  course  and  distance  from 
there  to  the  point  12  miles  east  of  Hinchinbroke  Rock  are : 
distance,  945  miles ;  course,  181°,  true,  or  173°  by  compass. 

Therefore,  by  the  table  on  page  52,  the  quartermaster  gets 
the  new  course  S.JE.  by  compass,  at  10  :  07.  This  corre- 
sponds to  174°  by  compass,  or  182°  true  course;  and  at 
10 : 07,  when  the  course  was  changed,  the  patent  log  read 
75.3. 

Now  the  Sumner  line,  from  the  observation  at  9*  42ra  28s 
by  the  watch,  was  worked  by  Kelvin's  table ;  and  the  result 
was  : 

Sumner  point  is  in  lat.  39°  50'  N. ;  long.  73°  56'  W. ;  bearing  of 
Sumner  line  237°. 

It  is  necessary,  as  a  check,  to  ascertain  whether  this  Sum- 
ner line  passes  through  the  position  obtained  for  the  ship 
by  the  Barnegat  bearings.  Before  doing  this,  the  Sumner 
point  must  be  shifted  by  the  method  of  page  137,  to  allow  for 


A  NAVIGATOR'S  DAY  AT   SEA  147 

the  motion  of  the  yacht  between  9 : 42,  when  the  sextant 
observation  was  made,  and  9 : 57,  when  Barnegat  bore 
abeam.  The  difference  is  15  minutes,  and  in  that  time  the 
ship  moved  south  3.4  miles  by  the  patent  log  and  an  in- 
significant distance  west. 

Therefore  the  corrected  Sumner  data  are : 

Sumner  point  is  in  lat.  39°  46'.6  N. ;  long.  73°  56'  W. ;  bearing  of 
Sumner  line  237°. 

If  everything  fits,  this  Sumner  line  must  pass  through  the 
Barnegat  "fix"  of  the  yacht  in  lat.  39°  45'  N. ;  long.  73°  59' 
W.,  because  the  yacht  must  have  been  somewhere  on  the 
line. 

The  traverse  table  shows  that  the  bearing  of  a  line  passing 
the  Sumner  point  and  the  yacht's  position  is  235°,  differing 
only  2°  from  the  Sumner  line  bearing ;  so  this  check  is  satis- 
factory. But  a  better  way  to  check  this  matter  is  to  deter- 
mine the  yacht's  position  from  the  intersection  of  two  lines, 
one  of  which  is  the  Sumner  line,  and  the  other  the  beam  bear- 
ing of  Barnegat  Light.  This  can  be  done  by  the  method  of 
page  133.  The  data  of  the  problem  are  : 

Sumner  point :    lat.  39°  46'.6  N. 

long.  73°  56'  W. 
Line  bears  237° 
Barnegat  Light :  lat.  39°  46'  N. 

long.  74°    6'  W. 
Line  bears  98° 

We  shall  call  Barnegat  Light  S' ;  and  then  formula  (3), 
page  136,  gives,  for  the  two  bow  bearings : 

At  Sumner  point,  S,  237°  -  266°  =   29°. 
At  Barnegat,         S',  98°  -  266°  =  168°. 

For  these  two  bearings,  Table  14  gives  the  factor  0.74,  and 
the  yacht  is  placed  6  miles  from  Barnegat,  on  the  98°  bear- 
ing. The  bow-and-beam  observations  gave  5.5  miles,  so 
the  check  by  the  Sumner  line  is  excellent. 

It  remains  for  the  captain  to  utilize  the  azimuth  observa- 


148  NAVIGATION 

tion  made  at  9 : 45.  The  bearing  of  the  Simmer  line  was 
237°,  and  therefore  the  sun's  true  azimuth  was  147°.  The 
observed  azimuth,  by  pelorus  (p.  145),  was  137°.  The  com- 
pass error  was  therefore  -f-  10°.  The  variation  being  -  10°, 
the  deviation  by  formula  (1),  page  45,  is  D  =  10°  -  (  -10°)  = 
+  20°. 

On  page  143  we  found  that  the  deviation  table  made  this 
deviation  +  18° ;  so  that  the  table  appears  to  require  a 
correction  of  +2°.  The  captain  decides  not  to  correct 
the  table  for  the  present,  unless  later  azimuth  observations 
shall  confirm  it,  especially  as  the  sunrise  observation  showed 
the  adjuster's  results  to  be  correct.  Azimuth  observa- 
tions made  when  the  sun  is  high  in  the  sky  are  not  quite 
as  reliable  as  sunrise  ones.  Moreover,  the  observation  was 
made  at  9 : 45,  whereas  the  altitude  observation,  for  which 
the  true  azimuth  was  calculated  with  Kelvin's  table,  was 
made  at  9 : 42,  so  that  the  true  azimuth  must  have  been  in 
error  by  the  sun's  azimuth  change  in  three  minutes.  This 
could  have  been  avoided  by  giving  the  mate  orders  to  ob- 
serve the  azimuth  at  about  the  same  moment  when  the 
captain  took  the  altitude.  Or,  the  sun's  azimuth  change 
in  three  minutes  might  be  taken  from  the  azimuth  table,  and 
the  computed  true  azimuth  duly  corrected. 

At  11  the  log  read  88.7,  and  the  course  was  S.JE.  by  com- 
pass, or  182°,  true. 

At  about  11 :  30,  the  weather  showing  signs  of  becoming 
thick,  no  preparations  were  made  for  a  noon-sight  by  the 
method  of  page  86 ;  and  rather  than  take  the  risk  of  losing  his 
noon  observation  altogether,  the  captain  took  an  ex-me- 
ridian altitude  at  11*  42™  0s  by  his  watch;  log  was  98.5; 
the  sextant  reading  26°  55' ;  index  +  3' ;  height  of  eye  15 
ft. ;  C.  —  W.  was  now  4*  51W.42S ;  and  chronometer  slow  4*. 

The  observation  was  worked  by  Kelvin's  table,  and  gave 
the  Sumner  point  in  lat.  39°  20'  N. ;  long.  73°  40'  W. ;  bearing 
of  Sumner  line  86°.  Figure  21  is  a  rough  sketch  of  this  Sumner 
line.  It  is  very  nearly  horizontal ;  had  the  observation  been 


A  NAVIGATOR'S  DAY  AT   SEA 


149 


39°20' 


made  at  noon  precisely,  it  would  have  been  perfectly  hori- 
zontal. 

It  would  now  have  been  possible  to  move  up  the  Sumner 
line  observed  at  9 :  42,  and  obtain  an  intersection  to  fix  the 
position  of  the  yacht. 
But  this  did  not  seem 
necessary  to  the  cap- 
tain, because  of  the 
beam  bearing  obtained 
at  Barnegat  at  9  :  57, 
which  gave  a  good  fix. 

And  the  present 
Sumner  line  being  so 
nearly  horizontal,  it  is 
not  necessary  to  know 
the  longitude  very  ac- 
curately to  obtain  an 
exact  latitude.  The 
longitude  by  D.  R.  is 
sufficient,  and  it  is  73°  58'  W.  The  difference  between 
this  longitude  and  that  of  the  Sumner  point  (73°  40')  is 
18' ;  and  the  ship  at  L  (fig.  21)  bears  180°  +  86°  =  266° 
from  the  Sumner  point.  Table  2  gives  the  dep.  14.0  for 
long.  diff.  18',  in  lat.  39°.  And  for  course  266°,  dep.  14.0', 
we  find  in  Table  1,  lat.  diff.  I'.O,  so  the  yacht's  latitude  is  I' 
less  than  that  of  the  Sumner  point,  and  is  therefore  39°  19'. 
This  happens  to  be  in  exact  accord  with  the  D.  R.  latitude, 
which  was  also  39°  19'.  This  was  perfectly  satisfactory, 
and  the  captain  decided  to  carry  this  Sumner  line  forward 
for  an  intersection,  in  case  he  should  obtain  an  observation 
in  the  afternoon. 

At  12,  the  patent  log  read  102.6,  course  S. JE.,  182°  true  ; 
D.  R.  lat.  39°  15' ;  long.  73°  58' ;  distance  to  Watlings  Island 
918  miles. 

Had  the  yacht  been  on  a  course  other  than  almost  due 
south, .it  would  have  been  necessary  to  set  the  watch  and  the 


FIG.  21. —  Sumner  Line  from  ex-Meridian 
Observation. 


150  NAVIGATION 

cabin  clock  to  ship's  apparent  time.  In  fact,  some  naviga- 
tors set  their  watches  to  ship's  apparent  time  before  every 
observation  (p.  94) : 

at  1,  log  read  117.7,  misty, 
at  2,  log  read  133.0,  misty, 
at  3,  log  read  149.0  misty, 
at  4,  log  read  163.8,  clearing. 

At  4s  I2m  IS8  by  the  watch,  the  weather  having  cleared, 
the  altitude  of  the  sun  was  found  to  be  4°  38' ;  index  +  4' ; 
eye  15  ft. ;  C.  —  W.  4*  51m  50* ;  chronometer  slow  4s ;  log 
166.9.  Sun's  azimuth,  observed  by  the  mate  at  the  same 
time,  came  out  224°  by  compass. 

This  observation  was  worked  for  a  Sumner  line  by  the 
Kelvin  table,  and  gave : 

Position  of  Sumner  point  lat.  38°  6'  N. ;  long.  73°  49'  W. ;  bearing 
of  line  145°  ;  azimuth  of  sun  235°. 

The  Sumner  line  obtained  at  11*  42m  0'  was  brought  up  to 
the  time  of  the  present  observation  by  D.  R.  (p.  137),  giving  : 

position    of  1 1 : 42  Sumner 

point,  after  moving  it,  lat. 

38°  12' N.;  long.  73°  43' W.; 

bearing    of    the    line    86°. 

Both     lines    were    then 

sketched,  as  shown  in  Fig. 

22.  The  point  S  is  the 
FIG.  22.  — Rough  Sketch  of  Sumner  (moved)  Sumner  point  from 

the  11:42  observation,  S' 

that  from  the  4:12  observation.  The  intersection  point  L  is 
the  position  of  the  ship  at  4 :  12,  and  it  came  out  (p.  134)  : 
lat.  38°  11'  N. ;  long.  73°  43'  W.  The  position  brought  up 
by  D.  R.  from  11 :42  was :  lat.  38°  11' ;  long.  74°  1' ;  so  that 
there  has  been  an  easterly  set  of  the  current,  amounting  to 
7'  of  longitude  in  4J  hours.  The  sun's  true  azimuth  at 
4 : 12  was  235°,  from  the  Kelvin  table ;  and  the  pelorus 
observation  gave  224°.  The  compass  error  was  therefore 


A  NAVIGATOR'S  DAY  AT  SEA  151 

-fll°.  The  variation  being  --  10°,  the  deviation  must 
beZ)  =  11°  -  (  -  10°=)  +  21°.  The  deviation  table  made 
this  deviation  +  18°,  so  that  table  seems  to  require  a  correc- 
tion of  +3°.  The  pelorus  observation  of  9  : 45  gave  a  correc- 
tion of  +  2°  for  the  deviation  table ;  and  as  this  is  now 
apparently  confirmed,  the  captain  decides  to  examine  the 
chart  again,  before  finally  shaping  course  for  the  night,  to 
see  if  the  yacht  has  not  perhaps  moved  into  a  region  where 
the  variation  is  different  from  the  Sandy  Hook  variation  so 
far  used. 

At  5  the  log  read  182.0,  course  was  still  182°  true. 

The  captain  now  prepared  to  shape  the  course  for  the 
night,  and  to  change  his  course,  if  necessary,  at  6 : 00.  His 
first  step  was  to  obtain  the  D.  R.  position  at  6 : 00,  starting 
from  the  observed  position  at  4 : 12.  This  gave  position  at 
6  :  00,  by  D.  R. :  lat.  37°  41' ;  long.  73°  55'.  The  easterly 
current l  of  about  2'  per  hour  set  the  yacht  farther  east  about 
3'  between  4  : 12  and  6  : 00.  Therefore  he  took  the  D.  R. 
position  at  6  :  00  to  be  lat.  37°  41' ;  long.  73°  52'.  The  posi- 
tion of  the  point  of  destination,  12  miles  east  of  Watlings 
Island,  is  still :  lat.  23°  57' ;  long.  74°  15'.  The  true  course 
and  distance  to  that  point  from  the  yacht's  6  : 00  position  is 
therefore,  by  traverse  table  :  course  181  J° ;  dist.  824  miles. 

A  further  examination  of  the  track  chart  shows  that  the 
variation,  which  was  —  10°  at  Sandy  Hook,  is  now  —  8°. 
The  compass  error,  from  the  last  pelorus  observation, 
was  +  11°.  Consequently,  by  the  pelorus  observation,  the 
compass  course  for  the  night  should  be  181J0  —  11°  =  170?°, 
or  S.fE.  (see  the  Table  on  p.  52).  Furthermore,  the 
variation  being  now  —  8°  and  the  error  +  11°  makes  the 
deviation  D  =  E  -  V  =  +  11°  -  (-  8°)  =  +  19°.  The  com- 
pass adjuster's  deviation  of  -f  18°  is  therefore  vindicated, 
and  the  compass  course  S.JE.  can  be  set  for  the  night. 
At  6  the  log  read  197.2,  course  S.fE.,  or  182*°  true. 
1  Doubtless  the  Gulf  Stream, 


152  NAVIGATION 

In  conclusion,  the  captain  of  the  Nav  hopes  he  has  been 
able  to  make  his  imagined  proceedings  clear  enough  to  help 
the  young  navigator  in  planning  his  own  first  day's  work  at 
sea.  May  it  be  the  first  of  many  happy  and  successful  days. 
And  let  him  not  forget,  when  attempting  to  verify  the 
various  calculations  and  problems  of  the  Nav,  that  every 
observation  in  this  book  has  been  prepared  by  calculation, 
and  none  is  the  result  of  actual  sextant  observing.  Should 
inconsistencies  or  errors  be  found  by  any  young  navigator,  it 
is  hoped  that  he  will  make  them  known  so  that  they  may  be 
corrected,  in  case  the  Nav  shall  be  required  to  make  another 
voyage  in  a  second  edition. 


LIST   OF   TABLES 

1.  Traverse  Table;   explained  on  pages  10  and  19;   and  its 

use  in  the  Sumner  method  on  pages  113,  135 154 

2.  Conversion   of  longitude  difference   and  departure;    ex- 

plained on  page  16 168 

3.  Number  logarithms ;   explained  on  page  23 178 

4.  Trigonometric  logarithms ;  explained  on  page  31 196 

5.  Meridional  parts ;   explained  on  page  35 241 

6.  Sextant  Correction  Table ;   explained  on  page  72 247 

7.  Dip  correction ;   explained  on  page  73 247 

8.  Conversion  of  hours  and  minutes  into  decimals  of  a  day ; 

explained  on  page  80 248 

9.  Conversion  of  degrees  and  minutes  of  longitude  and  hours 

and  minutes  of  time 249 

10.  Haversines ;   explained  on  page  99 250 

11.  Azimuth  Table ;   explained  on  page  113 284 

12.  Auxiliary  Azimuth  Table ;   explained  on  page  115 290 

13.  Kelvin's  Sumner  Line  Table ;   explained  on  page  126 292 

14.  Sumner  Intersection  Table;  .explained  on  page  135 318 


PUBLISHERS'   NOTE 

Table  3,  Number  Logarithms,  has  been  reprinted  from  "The 
Macmillan  Logarithmic  and  Trigonometric  Tables,"  New  York, 
1917. 


153 


154 


Table  1.    Traverse  Table 


1° 

2° 

i  Pt.  3° 

4° 

5° 

f  Pt.  6° 

7° 

(179°,  181°, 

(178°,  182° 

(177°,  183° 

(176°,  184° 

(175°,  185° 

(174°,  186° 

(173°,  187°, 

DlST 

359°) 

358°) 

357°) 

356°) 

355°) 

354°) 

353°) 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

Dep. 

1 

1.0 

0.0 

1.0 

0.0 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

2 

2.0 

0.0 

2.0 

0.1 

2.0 

0.1 

2.0 

0.1 

2.0 

0.2 

2.0 

0.2 

2.0 

0.2 

3 

3.0 

0.1 

3.0 

0.1 

3.0 

0.2 

3.0 

0.2 

3.0 

0.3 

3.0 

0.3 

3.0 

0.4 

4 

4.0 

0.1 

4.0 

0.1 

4.0 

0.2 

4.0 

0.3 

4.0 

0.3 

4.0 

0.4 

4.0 

0.5 

5 

5.0 

0.1 

5.0 

0.2 

5.0 

0.3 

5.0 

0.3 

5.0 

0.4 

5.0 

0.5 

5.0 

0.6 

6 

6.0 

0.1 

6.0 

0.2 

6.0 

0.3 

6.0 

0.4 

6.0 

0.5 

6.0 

0.6 

6.0 

0.7 

7 

7.0 

0.1 

7.0 

0.2 

7.0 

0.4 

7.0 

0.5 

7.0 

0.6 

7.0 

0.7 

6.9 

0.9 

8 

8.0 

0.1 

8.0 

0.3 

8.0 

0.4 

8.0 

0.6 

8.0 

0.7 

8.0 

0.8 

7.9 

1.0 

9 

9.0 

0.2 

9.0 

0.3 

9.0 

0.5 

9.0 

0.6 

9.0 

0.8 

9.0 

0.9 

8.9 

1.1 

10 

10.0 

0.2 

10.0 

0.3 

10.0 

0.5 

10.0 

0.7 

10.0 

0.9 

9.9 

1.0 

9.9 

1.2 

11 

11.0 

0.2 

11.0 

0.4 

11.0 

0.6 

11.0 

0.8 

11.0 

1.0 

10.9 

1.1 

10.9 

1.3 

12 

12.0 

0.2 

12.0 

0.4 

12.0 

0.6 

12.0 

0.8 

12.0 

1.0 

11.9 

1.3 

11.9 

1.5 

13 

13.0 

0.2 

13.0 

0.5 

13.0 

0.7 

13.0 

0.9 

13.0 

1.1 

12.9 

1.4 

12.9 

1.6 

14 

14.0 

0.2 

14.0 

0.5 

14.0 

0.7 

14.0 

1.0 

13.9 

1.2 

13.9 

1.5 

13.9 

1.7 

15 

15.0 

0.3 

15.0 

0.5 

15.0 

0.8 

15.0 

1.0 

14.9 

1.3 

14.9 

1.6 

14.9 

1.8 

16 

16.0 

0.3 

16.0 

0.6 

16.0 

0.8 

16.0 

1.1 

15.9 

1.4 

15.9 

1.7 

15.9 

1.9 

17 

17.0 

0.3 

17.0 

0.6 

17.0 

0.9 

17.0 

1.2 

16.9 

1.5 

16.9 

1.8 

16.9 

2.1 

18 

18.0 

0.3 

18.0 

0.6 

18.0 

0.9 

18.0 

1.3 

17.9 

1.6 

17.9 

1.9 

17.9 

2.2 

19 

19.0 

0.3 

19.0 

0.7 

19.0 

1.0 

19.0 

1.3 

18.9 

1.7 

18.9 

2.0 

18.9 

2.3 

20 

20.0 

0.3 

20.0 

0.7 

20.0 

1.0 

20.0 

1.4 

19.9 

1.7 

19.9 

2.1 

19.9 

2.4 

21 

21.0 

0.4 

21.0 

0.7 

21.0 

1.1 

20.9 

1.5 

20.9 

1.8 

20.9 

2.2 

20.8 

2.6 

22 

22.0 

0.4 

22.0 

0.8 

22.0 

1.2 

21.9 

1.5 

21.9 

1.9 

21.9 

2.3 

21.8 

2.7 

23 

23.0 

0.4 

23.0 

0.8 

23.0 

1.2 

22.9 

1.6 

22.9 

2.0 

22.9 

2.4 

22.8 

2.8 

24 

24.0 

0.4 

24.0 

0.8 

24.0 

1.3 

23.9 

1.7 

23.9 

2.1 

23.9 

2.5 

23.8 

2.9 

25 

25.0 

0.4 

25.0 

0.9 

25.0 

1.3 

24.9 

1.7 

24.9 

2.2 

24.9 

2.6 

24.8 

3.0 

26 

26.0 

0.5 

26.0 

0.9 

26.0 

1.4 

25.9 

1.8 

25.9 

2.3 

25.9 

2.7 

25.8 

3.2 

27 

27.0 

0.5 

27.0 

0.9 

27.0 

1.4 

26.9 

1.9 

26.9 

2.4 

26.9 

2.8 

26.8 

3.3 

28 

28.0 

0.5 

28.0 

1.0 

28.0 

1.5 

27.9 

2.0 

27.9 

2.4 

27.8 

2.9 

27.8 

3.4 

29 

29.0 

0.5 

29.0 

1.0 

29.0 

1  5 

28.9 

2.0 

28.9 

2.5 

28.8 

3.0 

28.8 

3.5 

30 

30.0 

0.5 

30.0 

1.0 

30.0 

i!e 

29.9 

2.1 

29.9 

2.6 

29.8 

3.1 

29.8 

3.7 

31 

31.0 

0'.5 

31.0 

1  1 

31.0 

1  6 

30.9 

?,?. 

30.9 

9  7 

30.8 

32 

30.8 

38 

32 

32.0 

0.6 

32.0 

1  i 

32.0 

1  7 

31.9 

9  9 

31.9 

•>  8 

31.8 

3  3 

31.8 

3  9 

33 

33.0 

0.6 

33.0 

1.2 

33.0 

1.7 

32.9 

2.3 

32.9 

2.9 

32.8 

3.4 

32.8 

4.0 

34 

34.0 

0.6 

34.0 

1.2 

34.0 

1.8 

33.9 

2.4 

33.9 

3.0 

33.8 

3.6 

33.7 

4.1 

35 

35.0 

0.6 

35.0 

1.2 

35.0 

1.8 

34.9 

2.4 

34.9 

3.1 

34.8 

3.7 

34.7 

4.3 

36 

36.0 

0.6 

36.0 

1.3 

36.0 

1.9 

35.9 

2.5 

35.9 

3.1 

35.8 

3.8 

35.7 

4.4 

37 

37.0 

0.6 

37.0 

1.3 

36.9 

1.9 

36.9 

2.6 

36.9 

3.2 

36.8 

3.9 

36.7 

4.5 

38 

38.0 

0.7 

38.0 

1.3 

37.9 

2.0 

37.9 

2.7 

37.9 

3.3 

37.8 

4.0 

37.7 

4.6 

39 

39.0 

0.7 

39.0 

1.4 

38.9 

2.0 

38.9 

2.7 

38.9 

3.4 

38.8 

4.1 

38.7 

4.8 

40 

40.0 

0.7 

40.0 

.4 

39.9 

2.1 

39.9 

2.8 

39.8 

3.5 

39.8 

4.2 

39.7 

4.9 

41 

41.0 

0.7 

41.0 

.4 

40.9 

2.1 

40.9 

2.9 

40.8 

3.6 

40.8 

4.3 

40.7 

5.0 

42 

42.0 

0.7 

42.0 

c 
.0 

41.9 

2.2 

41.9 

2.9 

41.8 

3.7 

41.8 

4.4 

41.7 

5.1 

43 

43.0 

0.8 

43.0 

.5 

42.9 

2.3 

42.9 

3.0 

42.8 

3.7 

42.8 

4.5 

42.7 

5.2 

44 

44.0 

0.8 

44.0 

.5 

43.9 

2.3 

43.9 

3.1 

43.8 

3.8 

43.8 

4.6 

43.7 

5.4 

45 

45.0 

0.8 

45.0 

.6 

44.9 

2.4 

44.9 

3.1 

44.8 

3.9 

44.8 

4.7 

44.7 

5.5 

46 

46.0 

0.8 

46.0 

.6 

45.9 

2.4 

45.9 

3.2 

45.8 

4.0 

45.7 

4.8 

45.7 

5.6 

47 

47.0 

0.8 

47.0 

.6 

46.9 

2.5 

46.9 

3.3 

46.8 

4.1 

46.7 

4.9 

46.6 

5.7 

48 

48.0 

0.8 

48.0 

.7 

47.9 

2.5 

47.9 

3.3 

47.8 

4.2 

47.7 

5.0 

47.6 

5.8 

49 

49.0 

0.9 

49.0 

.7 

48.9 

2.6 

48.9 

3.4 

48.8 

4.3 

48.7 

5.1 

48.6 

6.0 

50 

50.0 

0.9 

50.0 

.7 

49.9 

2.6 

49.9 

3.5 

49.8 

4.4 

49.7 

5.2 

49.6 

6.1 

100 

00.0 

1.7 

99.9 

3.5 

99.9 

5.2 

99.8 

7.0 

99.6 

8.7 

99.5 

10.5 

99.3 

12.2 

200 

200.0 

3.5 

199.9 

7.0 

199.7 

10.5 

199.5 

14.0 

199.2 

17.4 

198.9 

20.9 

198.5 

24.4 

300 

300.0 

5.2 

299.8 

10.5 

299.6 

15.7 

299.3 

20.9 

298.9 

26.1 

298.4 

31.4 

297.8 

36.6 

400 

399.9 

7.0 

399.8 

13.9 

399.4 

20.9 

399.0 

27.9 

398.5 

34.9 

397.8 

41.8 

397.0 

48.7 

500 

499.9 

8.8 

499.7 

17.4 

499.3 

26.2 

498.8 

34.8 

498.1 

43.6 

497.3 

52.3 

496.3 

61.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(91°,  269° 

(92°,  268°, 

(93°,  267°, 

(94°,  266°, 

(95°   265°, 

(96°,  264°, 

(97°,  263°, 

271°) 

272°) 

273°) 

274°) 

275°) 

276°) 

277°) 

89° 

88° 

7|Pt.87° 

86° 

85° 

7iPt.84° 

83° 

Table  1.    Traverse  Table 


1° 

2° 

1  Pt.  3° 

4° 

5° 

\  Pt.  6° 

7° 

DlST 

(179°,  181° 
359°) 

(178°,  182°, 
358°) 

(177°,  183°, 
357°) 

(176°,  184°, 
356°) 

(175°,  185°, 
355°) 

(174°,  186°, 
354°) 

(173°,  187°, 
353°) 

Lat. 

Dep 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

51.0 

0.9 

51.0 

1.8 

50.9 

2.7 

50.9 

3.6 

50.8 

4.4 

50.7 

5.3 

50.6 

6.2 

52 

52.0 

0.9 

52.0 

1.8 

51.9 

2.7 

51.9 

3.6 

51.8 

4.5 

51.7 

5.4 

51.6 

6.3 

53 

53.0 

0.9 

53.0 

1.8 

52.9 

2.8 

52.9 

3.7 

52.8 

4.6 

52.7 

5.5 

52.6 

6.5 

54 

54.0 

0.9 

54.0 

1.9 

53.9 

2.8 

53.9 

3.8 

53.8 

4.7 

53.7 

5.6 

53.6 

6.6 

55 

55.0 

1.0 

55.0 

1.9 

54.9 

2.9 

54.9 

3.8 

54.8 

4.8 

54.7 

5.7 

54.6 

6.7 

56 

56.0 

1.0 

56.0 

2.0 

55.9 

2.9 

55.9 

3.9 

55.8 

4.9 

55.7 

5.9 

55.6 

6.8 

57 

57.0 

1.0 

57.0 

2.0 

56.9 

3.0 

56.9 

4.0 

56.8 

5.0 

56.7 

6.0 

56.6 

6.9 

58 

58.0 

1.0 

58.0 

2.0 

57.9 

3.0 

57.9 

4.0 

57.8 

5.1 

57.7 

6.1 

57.6 

7.1 

59 

59.0 

1.0 

59.0 

2.1 

58.9 

3.1 

58.9 

4.1 

58.8 

5.1 

58.7 

6.2 

58.6 

7.2 

60 

60.0 

1.0 

60.0 

2.1 

59.9 

3.1 

59.9 

4.2 

59.8 

5.2 

59.7 

6.3 

59.6 

7.3 

61 

61.0 

1.1 

61.0 

2.1 

60.9 

3.2 

60.9 

4.3 

60.8 

5.3 

60.7 

6.4 

60.5 

7.4 

62 

62.0 

1.1 

62.0 

2.2 

61.9 

3.2 

61.8 

4.3 

61.8 

5.4 

61.7 

6.5 

61.5 

7.6 

63 

63.0 

1.1 

63.0 

2.2 

62.9 

3.3 

62.8 

4.4 

62.8 

5.5 

62.7 

6.6 

62.5 

7.7 

64 

64.0 

1.1 

64.0 

2.2 

63.9 

3.3 

63.8 

4.5 

63.8 

5.6 

63.6 

6.7 

63.5 

7.8 

65 

65.0 

1.1 

65.0 

2.3 

64.9 

3.4 

64.8 

4.5 

64.8 

5.7 

64.6 

6.8 

64.5 

7.9 

66 

66.0 

1.2 

66.0 

2.3 

65.9 

3.5 

65.8 

4.6 

65.7 

5.8 

65.6 

6.9 

65.5 

8:0 

67 

67.0 

1.2 

67.0 

2.3 

66.9 

3.5 

66.8 

4.7 

66.7 

5.8 

66.6 

7.0 

66.5 

8.2 

68 

68.0 

1.2 

68.0 

2.4 

67.9 

3.6 

67.8 

4.7 

67.7 

5.9 

67.6 

7.1 

67.5 

8.3 

69 

69.0 

1.2 

69.0 

2.4 

68.9 

3.6 

68.8 

4.8 

68.7 

6.0 

68.6 

7.2 

68.5 

8.4 

70 

70.0 

1.2 

70.0 

2.4 

69.9 

3.7 

69.8 

4.9 

69.7 

6.1 

69.6 

7.3 

69.5 

8.5 

71 

71.0 

1.2 

71.0 

2.5 

70.9 

3.7 

70.8 

5.0 

70.7 

6.2 

70.6 

7.4 

70.5 

&7 

72 

72.0 

1.3 

72.0 

2.5 

71.9 

3.8 

71.8 

5.0 

71.7 

6.3 

71.6 

7.5 

71.5 

8.8 

73 

73.0 

1.3 

73.0 

2.5 

72.9 

3.8 

72.8 

5.1 

72.7 

6.4 

72.6 

7.6 

72.5 

8.9 

74 

74.0 

1.3 

74.0 

2.6 

73.9 

3.9 

73.8 

5.2 

73.7 

6.4 

73.6 

7.7 

73.4 

9.0 

75 

75.0 

1.3 

75.0 

2.6 

74.9 

3.9 

74.8 

5.2 

74.7 

6.5 

74.6 

7.8 

74.4 

9.1 

76 

76.0 

1.3 

76.0 

2.7 

75.9 

4.0 

75.8 

5.3 

75.7 

6.6 

75.6 

7.9 

75.4 

9.3 

77 

77.0 

1.3 

77.0 

2.7 

76.9 

4.0 

76.8 

5.4 

76.7 

6.7 

76.6 

8.0 

76.4 

9.4 

78 

78.0 

1.4 

78.0 

2.7 

77.9 

4.1 

77.8 

5.4 

77.7 

6.8 

77.6 

8.2 

77.4 

9.5 

79 

79.0 

1.4 

79.0 

2.8 

78.9 

4.1 

78.8 

5.5 

78.7 

6.9 

78.6 

8.3 

78.4 

9.6 

80 

80.0 

1.4 

80.0 

2.8 

79.9 

4.2 

79.8 

5.6 

79.7 

7.0 

79.6 

8.4 

79.4 

9.7 

81 

81.0 

1.4 

81.0 

2.8 

80.9 

4.2 

80.8 

5.7 

80.7 

7.1 

80.6 

8.5 

80.4 

9.9 

82 

82.0 

1.4 

82.0 

2.9 

81.9 

4.3 

81.8 

5.7 

81.7 

7.1 

81.6 

8.6 

81.4 

10.0 

83 

83.0 

1.4 

82.9 

2.9 

82.9 

4.3 

82.8 

5.8 

82.7 

7.2 

82.5 

8.7 

82.4 

10.1 

84 

84.0 

1.5 

83.9 

2.9 

83.9 

4.4 

83.8 

5.9 

83.7 

7.3 

83.5 

8.8 

83.4 

10.2 

85 

85.0 

1.5 

84.9 

3.0 

84.9 

4.4 

84.8 

5.9 

84.7 

7.4 

84.5 

8.9 

84.4 

10.4 

86 

86.0 

1.5 

85.9 

3.0 

85.9 

4.5 

85.8 

6.0 

85.7 

7.5 

85.5 

9.0 

85.4 

10.5 

87 

87.0 

1.5 

86.9 

3.0 

86.9 

4.6 

86.8 

6.1 

86.7 

7.6 

86.5 

9.1 

86.4 

10.6 

88 

88.0 

1.5 

87.9 

3.1 

87.9 

4.6 

87.8 

6.1 

87.7 

7.7 

87.5 

9.2 

87.3 

10.7 

89 

89.0 

1.6 

88.9 

3.1 

88.9 

4.7 

88.8 

6.2 

88.7 

7.8 

88.5 

9.3 

88.3 

10.8 

90 

90.0 

1.6 

89.9 

3.1 

89.9 

4.7 

89.8 

6.3 

89.7 

7.8 

89.5 

9.4 

89.3 

11.0 

"  91 

91.0 

1.6 

90.9 

3.2 

90.9 

4.8 

90.8 

6.3 

90.7 

7.9 

90.5 

9.5 

90.3 

11.1 

92 

92.0 

1.6 

91.9 

3.2 

91.9 

4.8 

91.8 

6.4 

91.6 

8.0 

91.5 

9.6 

91.3 

11.2 

93 

93.0 

1.6 

92.9 

3.2 

92.9 

4.9 

92.8 

6.5 

92.6 

8.1 

92.5 

9.7 

92.3 

11.3 

94 

94.0 

1.6 

93.9 

3.3 

93.9 

4.9 

93.8 

6.6 

93.6 

8.2 

93.5 

9.8 

93.3 

11.5 

95 

95.0 

1.7 

94.9 

3.3 

94.9 

5.0 

94.8 

6.6 

94.6 

8.3 

94.5 

9.9 

94.3 

11.6 

96 

96.0 

1.7 

95.9 

3.4 

95.9 

5.0 

95.8 

6.7 

95.6 

8.4 

95.5 

10.0 

95.3 

11.7 

97 

97.0 

1.7 

96.9 

3.4 

96.9 

5.1 

96.8 

6.8 

96.6 

8.5 

96.5 

10.1 

96.3 

11.8 

98 

98.0 

1.7 

97.9 

3.4 

97.9 

5.1 

97.8 

6.8 

97.6 

8.5 

97.5 

10.2 

97.3 

11.9 

99 

99.0 

1.7 

98.9 

3.5 

98.9 

5.2 

98.8 

6.9 

98.6 

8.6 

98.5 

10.3 

98.3 

12.1 

100 

100.0 

1.7 

99.9 

3.5 

99.9 

5.2 

99.8 

7.0 

99.6 

8.7 

99.5 

10.5 

99.3 

12.2 

600 

599.9 

10.5 

599.6 

20.9 

599.2 

31.4 

598.6 

41.9 

597.7 

52.3 

596.7 

62.7 

595.5 

73.1 

700 

699.8 

12.2 

699.5 

24.4 

699.0 

36.6 

698.2 

48.8 

697.2 

61.0 

696.1 

73.2 

694.9 

85.3 

800 

799.8 

14.0 

799.5 

27.9 

798.9 

41.9 

798.0 

55.8 

796.9 

69.7 

795.6 

83.6 

794.1 

97.5 

900 

899.7 

15.7 

899.3 

31.4 

898.6 

47.1 

897.6 

62.8 

896.4 

78.4 

895.0 

94.1 

893.3 

109.6 

Dep. 

LaT 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(91°,  269°, 

(92°,  268° 

(93°,  267°, 

(94°,  266°, 

(95°,  265°, 

(96°,  264°, 

(97°,  263°, 

271°) 

272°) 

273°) 

274°) 

275°) 

276°) 

277°) 

89° 

88° 

7f  Pt.  87° 

86° 

85° 

7§  Pt.  84° 

83° 

156 


Table  1.    Traverse  Table 


f  Pt.  8° 

9° 

10° 

1  Pt.  11° 

12° 

13° 

1  1  Pt.  14° 

DlST. 

(172°,  188°, 
352°) 

(171°,  189°, 
351°) 

(170°,  190°, 
350°) 

(169°,  191°, 
349°) 

(168°,  192°, 
348°) 

(167°,  193°, 
347°) 

(166°,  194°, 
346°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.0 

0.1 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

2 

2.0 

0.3 

2.0 

0.3 

2.0 

0.3 

2.0 

0.4 

2.0 

0.4 

1.9 

0.4 

1.9 

0.5 

3 

3.0 

0.4 

3.0 

0.5 

3.0 

0.5 

2.9 

0.6 

.2.9 

0.6 

2.9 

0.7 

2.9 

0.7 

4 

4.0 

0.6 

4.0 

0.6 

3.9 

0.7 

3.9 

0.8 

3.9 

0.8 

3.9 

0.9 

3.9 

1.0 

5 

5.0 

0.7 

4.9 

0.8 

4.9 

0.9 

4.9 

1.0 

4.9 

.0 

4.9 

1.1 

4.9 

1.2 

6 

5.9 

0.8 

5.9 

0.9 

5.9 

.0 

5.9 

1.1 

5.9 

2 

5.8 

1.3 

5.8 

1.5 

7 

6.9 

.0 

6.9 

1.1 

6.9 

.2 

6.9 

1.3 

6.8 

'.5 

6.8 

1.6 

6.8 

1.7 

8 

7.9 

.1 

7.9 

1.3 

7.9 

.4 

7.9 

1.5 

7.8 

.7 

7.8 

1.8 

7.8 

1.9 

9 

8.9 

.3 

8.9 

1.4 

8.9 

.6 

8.8 

1.7 

8.8 

.9 

8.8 

2.0 

8.7 

2.2 

10 

9.9 

.4 

9.9 

1.6 

9.8 

.7 

9.8 

1.9 

9.8 

2.1 

9.7 

2.2 

9.7 

2.4 

11 

10.9 

.5 

10.9 

1.7 

10.8 

1.9 

10.8 

2.1 

10.8 

2.3 

10.7 

2.5 

10.7 

2.7 

12 

11.9 

.7 

11.9 

1.9 

11.8 

2.1 

11.8 

2.3 

11.7 

2.5 

11.7 

2.7 

11.6 

2.9 

13 

12.9 

.8 

12.8 

2.0 

12.8 

2.3 

12.8 

2.5 

12.7 

2.7 

12.7 

2.9 

12.6 

3.1 

14 

13.9 

1.9 

13.8 

2.2 

13.8 

2.4 

13.7 

2.7 

13.7 

2.9 

13.6 

3J 

13.6 

3.4 

15 

14.9 

2.1 

14.8 

2.3 

14.8 

2.6 

14.7 

2.9 

14.7 

3.1 

14.6 

3.4 

14.6 

3.6 

16 

15.8 

2.2 

15.8 

2.5 

15.8 

2.8 

15.7 

3.1 

15.7 

3.3 

15.6 

3.6 

15.5 

3.9 

17 

16.8 

2.4 

16.8 

2.7 

16.7 

3.0 

16.7 

3.2 

16.6 

3.5 

16.6 

3.8 

16.5 

4.1 

18 

17.8 

2.5 

17.8 

2.9 

17.7 

3.1 

17.7 

3.4 

17.6 

3.7 

17.5 

4.0 

17.5 

4.4 

19 

18.8 

2.6 

18.8 

3.0 

18.7 

3.3 

18.7 

3.6 

18.6 

4.0 

18.5 

4.3 

18.4 

4.6 

20 

19.8 

2.8 

19.8 

3.1 

19.7 

3.5 

19.6 

3.8 

19.6 

4.2 

19.5 

4.5 

19.4 

4.8 

21 

20.8 

2.9 

20.7 

3.3 

20.7 

3.6 

20.6 

4.0 

20.5 

4.4 

20.5 

4.7 

20.4 

5.1 

22 

21.8 

3.1 

21.7 

3.4 

21.7 

3.8 

21.6 

4.2 

21.5 

4.6 

21.4 

4.9 

21.3 

5.3 

23 

22.8 

3.2 

22.7 

3.6 

22.7 

4.0 

22.6 

4.4 

22.5 

4.8 

22.4 

5.2 

22.3 

5.6 

24 

23.8 

3.3 

23.7 

3.8 

23.6 

4.2 

23.6 

4.6 

23.5 

5.0 

23.4 

5.4 

23.3 

5.8 

25 

24.8 

3.5 

24.7 

3.9 

24.6 

4.3 

24.5 

4.8 

24.5 

5.2 

24.4 

5.6 

24.3 

6.0 

26 

25.7 

3.6 

25.7 

4.1 

25.6 

4.5 

25.5 

5.0 

25.4 

5.4 

25.3 

5.8 

25.2 

6.3 

27 

26,7 

3.8 

26.7 

4.2 

26.6 

4.7 

26.5 

5.2 

26.4 

5.6 

26.3 

6.1 

26.2 

6.5 

28 

27.7 

3.9 

27.7 

4.4 

27.6 

4.9 

27.5 

5.3 

27.4 

5.8 

27.3 

6.3 

27.2 

6.8 

29 

28.7 

4.0 

28.6 

4.5 

28.6 

5.0 

28.5 

5.5 

28.4 

6.0 

28.3 

6.5 

28.1 

7.0 

30 

29.7 

4.2 

29.6 

4.7 

29.5 

5.2 

29.4 

5.7 

29.3 

6.2 

29.2 

6.7 

29.1 

7.3 

31 

30.7 

4.3 

30.6 

4.8 

30.5 

5.4 

30.4 

5.9 

30.3 

6.4 

30.2 

7.0 

30.1 

7.5 

32 

31.7 

4.5 

31.6 

5.0 

31.5 

5.6 

31.4 

6.1 

31.3 

6.7 

31.2 

7.2 

31.0 

7.7 

33 

32.7 

4.6 

32.6 

5.2 

32.5 

5.7 

32.4 

6.3 

32.3 

6.9 

32.2 

7.4 

32.0 

8.0 

34 

33.7 

4.7 

33.6 

5.3 

33.5 

5.9 

33.4 

6.5 

33.3 

7.1 

33.1 

7.6 

33.0 

8.2 

35 

34.7 

4.9 

34.6 

5.5 

34.5 

6.1 

34.4 

6.7 

34.2 

7.3 

34.1 

7.9 

34.0 

8.5 

36 

35.6 

5.0 

35.6 

5.6 

35.5 

6.3 

35.3 

6.9 

35.2 

7.5 

35.1 

8.1 

34.9 

8.7 

37 

36.6 

5.1 

36.5 

5.8 

36.4 

6.4 

36.3 

7.1 

36.2 

7.7 

36.1 

8.3 

35.9 

9.0 

38 

37.6 

5.3 

37.5 

5.9 

37.4 

6.6 

37.3 

7.3 

37.2 

7.9 

37.0 

8.5 

36.9 

9.2 

39 

38.6 

5.4 

38.5 

6.1 

38.4 

6.8 

38.3 

7.4 

38.1 

8.1 

38.0 

8.8 

37.8 

9.4 

40 

39.6 

5.6 

39.5 

6.3 

39.4 

6.9 

39.3 

7.6 

39.1 

8.3 

39.0 

9.0 

38.8 

9.7 

41 

40.6 

5.7 

40.5 

6.4 

40.4 

7.1 

40.2 

7.8 

40.1 

8.5 

39.9 

9.2 

39.8 

9.9 

42 

41.6 

5.8 

41.5 

6.6 

41.4 

7.3 

41.2 

8.0 

41.1 

8.7 

40.9 

9.4 

40.8 

10.2 

43 

42.6 

6.0 

42.5 

6.7 

42.3 

7.5 

42.2 

8.2 

42.1 

8.9 

41.9 

9.7 

41.7 

10.4 

44 

43.6 

6.1 

43.5 

6.9 

43.3 

7.6 

43.2 

8.4 

43.0 

9.1 

42.9 

9.9 

42.7 

10.6 

45 

44.6 

6.3 

44.4 

7.0 

44.3 

7.8 

44.2 

8.6 

44.0 

9.4 

43.8 

10.1 

43.7 

10.9 

46 

45.6 

6.4 

45.4 

7.2 

45.3 

8.0 

45.2 

8.8 

45.0 

9.6 

44.8 

10.3 

44.6 

11.1 

47 

46.5 

6.5 

46.4 

7.4 

46.3 

8.2 

46.1 

9.0 

46.0 

9.8 

45.8 

10.6 

45.6 

11.4 

48 

47.5 

6.7 

47.4 

7.5 

47.3 

8.3 

47.1 

9.2 

47.0 

10.0 

46.8 

10.8 

46.6 

11.6 

49 

48.5 

6.8 

48.4 

7.7 

48.3 

8.5 

48.1 

9.3 

47.9 

10.2 

47.7 

11.0 

47.5 

11.9 

50 

49.5 

7.0 

49.4 

7.8 

49.2 

8.7 

49.1 

9.5 

48.9 

10.4 

48.7 

11.2 

48.5 

12.1 

100 

99.0 

13.9 

98.8 

15.6 

98.5 

17.4 

98.2 

19.1 

97.8 

20.8 

97.4 

22.5 

97.0 

24.2 

200 

198.1 

27.8 

197.5 

31.3 

197.0 

34.7 

196.3 

38.2 

195.6 

41.6 

194.9 

45.0 

194.1 

48.4 

300 

297.1 

41.8 

296.3 

46.9 

295.4 

52.1 

294.5 

57.2 

293.4 

62.4 

292.3 

67.5 

291.1 

72.6 

400 

396.1 

55.7 

395.1 

62.6 

393.9 

69.5 

392.6 

76.3 

391.3 

83.1 

389.8 

90.0 

388.1 

96.7 

500 

495.1 

69.6 

493.8 

78.2 

492.4 

86.8 

490.8 

95.4 

489.1 

104.0 

487.2 

112.4 

485.1 

121.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(98°,  262°, 

(99°,  261°, 

(100°,  260°, 

(101°,  259°, 

(102°,  258°, 

(103°,  257°, 

(104°,  256°, 

278°) 

279°) 

280°) 

281°) 

282°) 

283°) 

284°) 

7i  Pt.  82° 

81° 

80° 

7  Pt.  79° 

78° 

77° 

6  f  Pt.  76° 

The  1-Pt.  or  11°  Courses  are  :  N.  by  E.,  N.  by  W.,  S.  by  E.,  S.  by  W. 


Table  1.    Traverse  Table 


f  Pt.  8° 

9° 

10° 

1  Pt.  11° 

12° 

13° 

UPt.  14° 

(172°,  188°, 

(171°,  189°. 

(170°,  190°, 

(169°,  191°, 

(168°,  192°, 

(167°,  193°, 

(166°,  194°, 

DlST. 

352°) 

351°) 

350°) 

349°) 

348°) 

347°) 

346°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

50.5 

7.1 

50.4 

8.0 

50.2 

8.9 

50.1 

9.7 

49.9 

10.6 

49.7 

11.5 

49.5 

12.3 

52 

51.5 

7.2 

51.4 

8.1 

51.2 

9.0 

51.0 

9.9 

50.9 

10.8 

50.7 

11.7 

50.5 

12.6 

53 

52.5 

7.4 

52.3 

8.3 

52.2 

9.2 

52.0 

10.1 

51.8 

11.0 

51.6 

11.9 

51.4 

12.8 

54 

53.5 

7.5 

53.3 

8.4 

53.2 

9.4 

53.0 

10.3 

52.8 

11.2 

52.6 

12.1 

52.4 

13.1 

55 

54.5 

7.7 

54.3 

8.6 

54.2 

9.6 

54.0 

10.5 

53.8 

11.4 

53.6 

12.4 

53.4 

13.3 

56 

55.5 

7.8 

55.3 

8.8 

55.1 

9.7 

55.0 

10.7 

54.8 

11.6 

54.6 

12.6 

54.3 

13.5 

57 

56.4 

7.9 

56.3 

8.9 

56.1 

9.9 

56.0 

10.9 

55.8 

11.9 

55.5 

12.8 

55.3 

13.8 

:,s 

57.4 

8.1 

57.3 

9.1 

57.1 

10.1 

56.9 

11.1 

56.7 

12.1 

56.5 

13.0 

56.3 

14.0 

59 

58.4 

8.2 

58.3 

9.2 

58.1 

10.2 

57.9 

11.3 

57.7 

12.3 

57.5 

13.3 

57.2 

14.3 

60 

59.4 

8.4 

59.3 

9.4 

59.1 

10.4 

58.9 

11.4 

58.7 

12.5 

58.5 

13.5 

58.2 

14.5 

61 

60.4 

8.5 

60.2 

9.5 

60.1 

10.6 

59.9 

11.6 

59.7 

12.7 

59.4 

13.7 

59.2 

14.8 

62 

61.4 

8.6 

61.2 

9.7 

61.1 

10.8 

60.9 

11.8 

60.6 

12.9 

60.4 

13.9 

60.2 

15.0 

63 

62.4 

8.8 

62.2 

9.9 

62.0 

10.9 

61.8 

12.0 

61.6 

13.1 

61.4 

14.2 

61.1 

15.2 

64 

63.4 

8.9 

63.2 

10.0 

63.0 

11.1 

62.8 

12.2 

62.6 

13.3 

62.4 

14.4 

62.1 

15.5 

65 

64.4 

9.0 

64.2 

10.2 

64.0 

11.3 

63.8 

12.4 

63.6 

13.5 

63.3 

14.6 

63.1 

15.7 

66 

65.4 

9.2 

65.2 

10.3 

65.0 

11.5 

64.8 

12.6 

64.6 

13.7 

64.3 

14.8 

64.0 

16.0 

67 

66.3 

9.3 

66.2 

10.5 

66.0 

11.6 

65.8 

12.8 

65.5 

13.9 

65.3 

15.1 

65.0 

16.2 

68 

67.3 

9.5 

67.2 

10.6 

67.0 

11.8 

66.8 

13.0 

66.5 

14.1 

66.3 

15.3 

66.0 

16.5 

69 

68.3 

9.6 

68.2 

10.8 

68.0 

12.0 

67.7 

13.2 

67.5 

14.3 

67.2 

15.5 

67.0 

16.7 

70 

69.3 

9.7 

69.1 

11.0 

68.9 

12.2 

68.7 

13.4 

68.5 

14.6 

68.2 

15.7 

67.9 

16.9 

71 

70.3 

9.9 

70.1 

11.1 

69.9 

12.3 

69.7 

13.5 

69.4 

14.8 

69.2 

16.0 

68.9 

17.2 

72 

71.3 

10.0 

71.1 

11.3 

70.9 

12.5 

70.7 

13.7 

70.4 

15.0 

70.2 

16.2 

69.9 

17.4 

73 

72.3 

10.2 

72.1 

11.4 

71.9 

12.7 

71.7 

13.9 

71.4 

15.2 

71.1 

16.4 

70.8 

17.7 

74 

73.3 

10.3 

73.1 

11.6 

72.9 

12.8 

72.6 

14.1 

72.4 

15.4 

72.1 

16.6 

71.8 

17.9 

75 

74.3 

10.4 

74.1 

11.7 

73.9 

13.0 

73.6 

14.3 

73.4 

15.6 

73.1 

16.9 

72.8 

18.1 

76 

75.3 

10.6 

75.1 

11.9 

74.8 

13.2 

74.6 

14.5 

74.3 

15.8 

74.1 

17.1 

73.7 

18.4 

77 

76.3 

10.7 

76.1 

12.0 

75.8 

13.4 

75.6 

14.7 

75.3 

16.0 

75.0 

17.3 

74.7 

18.6 

78 

77.2 

10.9 

77.0 

12.2 

76.8 

13.5 

76.6 

14.9 

76.3 

16.2 

76.0 

17.5 

75.7 

18.9 

79 

78.2 

11.0 

78.0 

12.4 

77.8 

13.7 

77.5 

15.1 

77.3 

16.4 

77.0 

17.8 

76.7 

19.1 

80 

79.2 

11.1 

79.0 

12.5 

78.8 

13.9 

78.5 

15.3 

78.3 

16.6 

77.9 

18.0 

77.6 

19.4 

81 

80.2 

11.3 

80.0 

12.7 

79.8 

14.1 

79.5 

15.5 

79.2 

16.8 

78.9 

18.2 

78.6 

19.6 

82 

81.2 

11.4 

81.0 

12.8 

80.8 

14.2 

80.5 

15.6 

80.2 

17.0 

79.9 

18.4 

79.6 

19.8 

83 

82.2 

11.6 

82.0 

13.0 

81.7 

14.4 

81.5 

15.8 

81.2 

17.3 

80.9 

18.7 

80.5 

20.1 

84 

83.2 

11.7 

83.0 

13.1 

82.7 

14.6 

82.5 

16.0 

82.2 

17.5 

81.8 

18.9 

81.5 

20.3 

85 

84.2 

11.8 

S4.0 

13.3 

83.7 

14.8 

83.4 

16.2 

83.1 

17.7 

82.8 

19.1 

82.5 

20.6 

86 

85.2 

12.0 

84.9 

13.5 

84.7 

14.9 

84.4 

16.4 

84.1 

17.9 

83.8 

19.3 

83.4 

20.8 

87 

86.2 

12.1 

85.9 

13.6 

85.7 

15.1 

85.4 

16.6 

85.1 

18.1 

84.8 

19.6 

84.4 

21.0 

88 

87.1 

12.2 

86.9 

13.8 

86.7 

15.3 

86.4 

16.8 

86.1 

18.3 

85.7 

19.8 

85.4 

21.3 

89 

88.1 

12.4 

87.9 

13.9 

87.6 

15.5 

87.4 

17.0 

87.1 

18.5 

86.7 

20.0 

86.4 

21.5 

90 

89.1 

12.5 

88.9 

14.1 

88.6 

15.6 

88.3 

17.2 

88.0 

18.7 

87.7 

20.2 

87.3 

21.8 

91 

90.1 

12.7 

89.9 

14.2 

89.6 

15.8 

89.3 

17.4 

89.0 

18.9 

88.7 

20.5 

88.3 

22.0 

92 

91.1 

12.8 

90.9 

14.4 

90.6 

16.0 

90.3 

17.6 

90.0 

19.1 

89.6 

20.7 

89.3 

22.3 

93 

92.1 

12.9 

91.9 

14.5 

91.6 

16.1 

91.3 

17.7 

91.0 

19.3 

90.6 

20.9 

90.2 

22.5 

94 

93.1 

13.1 

92.8 

14.7 

92.6 

16.3 

92.3 

17.9 

91.9 

19.5 

91.6 

21.1 

91.2 

22.7 

95 

94.1 

13.2 

93.8 

14.9 

93.6 

16.5 

93.3 

18.1 

92.9 

19.8 

92.6 

21.4 

92.2 

23.0 

96 

95.1 

13.4 

94.8 

15.0 

94.5 

16.7 

94.2 

18.3 

93.9 

20.0 

93.5 

21.6 

93.1 

23.2 

97 

96.1 

13.5 

95.8 

15.2 

95.5 

16.8 

95.2 

18.5 

94.9 

20.2 

94.5 

21.8 

94.1 

23.5 

98 

97.0 

13.6 

96.8 

15.3 

96.5 

17.0 

96.2 

18.7 

95.9 

20.4 

95.5 

22.0 

95.1 

23.7 

99 

98.0 

13.8 

97.8 

15.5 

97.5 

17.2 

97.2 

18.9 

96.8 

20.6 

96.5 

22.3 

96.1 

24.0 

100 

99.0 

13.9 

98.8 

15.6 

98.5 

17.4 

98.2 

19.1 

97.8 

20.8 

97.4 

22.5 

97.0 

24.2 

BOO 

594.2 

83.5 

592.6 

93.8 

590.9 

104.2 

589.0 

114.5 

586.9 

124.7 

584.6 

135.0 

582.2 

145.1 

700 

693.3 

97.4 

691.3 

109.4 

689.5 

121.5 

687.1 

133.6 

684.7 

145.5 

682.1 

157.5 

679.2 

169.3 

son 

792.3 

111.4 

790.2 

125.1 

787.9 

139.0 

785.2 

152.6 

782.5 

166.3 

779.4 

180.0 

776.2 

193.6 

900 

891.3 

125.2 

888.8 

140.8 

886.3 

156.3 

883.3 

171.7 

880.2 

187.1 

876.8 

202.4 

873.2 

217.7 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(98°,  262°, 

(99°,  261°, 

(100°,  260°, 

(101°,  259°, 

(102°,  258°, 

(103°,  257°, 

(104°,  256°, 

278°) 

279°) 

280°) 

281°) 

282°) 

283°) 

284°) 

7-1  Pt.  82° 

81° 

80° 

7  Pt.  79° 

78° 

77° 

6f  Pt.  76° 

The  7-Pt.  or  79°  Courses  are :  E.  by  N.,  W.  by  N.,  E.  by  S.,  W.  by  S. 


158 


Table  1.    Traverse  Table 


15° 

16° 

UPt.l7° 

18° 

19° 

If  Pt.  20° 

(165°,  195°, 

(164°,  196°, 

(163°,  197°, 

(162°,  198°, 

(161°,  199°, 

(160°,  200°, 

DlST. 

345°) 

344°) 

343°) 

342°) 

341°) 

340°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.0 

0.3 

1.0 

0.3 

1.0 

0.3 

1.0 

0.3 

0.9 

0.3 

0.9 

0.3 

2 

1.9 

0.5 

1.9 

0.6 

1.9 

0.6 

1.9 

0.6 

1.9 

0.7 

1.9 

0.7 

3 

2.9 

0.8 

2.9 

0.8 

2.9 

0.9 

2.9 

0.9 

2.8 

1.0 

2.8 

1.0 

4 

3.9 

1.0 

3.8 

1.1 

3.8 

1.2 

3.8 

1.2 

3.8 

1.3 

3.8 

1.4 

5 

4.8 

1.3 

4.8 

1.4 

4.8 

1.5 

4.8 

1.5 

4.7 

1.6 

4.7 

1.7 

6 

5.8 

1.6 

5.8 

1.7 

5.7 

1.8 

5.7 

1.9 

5.7 

2.0 

5.6 

2.1 

7 

6.8 

1.8 

6.7 

1.9 

6.7 

2.0 

6.7 

2.2 

6.6 

2.3 

6.6 

2.4 

8 

7.7 

2.1 

7.7 

2.2 

7.7 

2.3 

.  7.6 

2.5 

7.6 

2.6 

7.5 

2.7 

9 

8.7 

2.3 

8.7 

2.5 

8.6 

2.6 

8.6 

2.8 

8.5 

2.9 

8.5 

3.1 

10 

9.7 

2.6 

9.6 

2.8 

9.6 

2.9 

9.5 

3.1 

9.5 

3.3 

9.4 

3.4 

11 

10.6 

2.8 

10.6 

3.0 

10.5 

3.2 

10.5 

3.4 

10.4 

3.6 

10.3 

3.8 

12 

11.6 

3.1 

11.5 

3.3 

11.5 

3.5 

11.4 

3.7 

11.3 

3.9 

11.3 

4.1 

13 

12.6 

3.4 

12.5 

3.6 

12.4 

3.8 

12.4 

4.0 

12.3 

4.2 

12.2 

4.4 

14 

13.5 

3.6 

13.5 

3.9 

13.4 

4.1 

13.3 

4.3 

13.2 

4.6 

13.2 

4.8 

15 

14.5 

3.9 

14.4 

4.1 

14.3 

4.4 

14.3 

4.6 

14.2 

4.9 

14.1 

5.1 

16 

15.5 

4.1 

15.4 

4.4 

15.3 

4.7 

15.2 

4.9 

15.1 

5.2 

15.0 

5.5 

17 

16.4 

4.4 

16.3 

4.7 

16.3 

5.0 

16.2 

5.3 

16.1 

5.5 

16.0 

5.8 

18 

17.4 

4.7 

17.3 

5.0 

17.2 

5.3 

17.1 

5.6 

17.0 

5.9 

16.9 

6.2 

19 

18.4 

4.9 

18.3 

5.2 

18.2 

5.6 

18.1 

5.9 

18.0 

6.2 

17.9 

6.5 

20 

19.3 

5.2 

19.2 

5.5 

19.1 

5.8 

19.0 

6.2 

18.9 

6.5 

18.8 

6.8 

21 

20.3 

5.4 

20.2 

5.8 

20.1 

6.1 

20.0 

6.5 

19.9 

6.8 

19.7 

7.2 

22 

21.3 

5.7 

21.1 

6.1 

21.0 

6.4 

20.9 

6.8 

20.8 

7.2 

20.7 

7.5 

23 

22.2 

6.0 

22.1 

6.3 

22.0 

6.7 

21.9 

7.1 

21.7 

7.5 

21.6 

7.9 

24 

23.2 

6.2 

23.1 

6.6 

23.0 

7.0 

22.8 

7.4 

22.7 

7.8 

22.6 

8.2 

25 

24.1 

6.5 

24.0 

6.9 

23.9 

7.3 

23.8 

7.7 

23.6 

8.1 

23.5 

8.6 

26 

25.1 

6.7 

25.0 

7.2 

24.9 

7.6 

24.7 

8.0 

24.6 

8.5 

24.4 

8.9 

27 

26.1 

7.0 

26.0 

7.4 

25.8 

7.9 

25.7 

8.3 

25.5 

8.8 

25.4 

9.2 

28 

27.0 

7.2 

26.9 

7.7 

26.8 

8.2 

26.6 

8.7 

26.5 

9.1 

26.3 

9.6 

29 

28.0 

7.5 

27.9 

8.0 

27.7 

8.5 

27.6 

9.0 

27.4 

9.4 

27.3 

9.9 

30 

29.0 

7.8 

28.8 

8.3 

28.7 

8.8 

28.5 

9.3 

28.4 

9.8 

28.2 

10.3 

31 

29.9 

8.0 

29.8 

8.5 

29.6 

9.1 

29.5 

9.6 

29.3 

10.1 

29.1 

10.6 

32 

30.9 

8.3 

30.8 

8.8 

30.6 

9.4 

30.4 

9.9 

30.3 

10.4 

30.1 

10.9 

33 

31.9 

8.5 

31.7 

9.1 

31.6 

9.6 

31.4 

10.2 

31.2 

10.7 

31.0 

11.3 

34 

32.8 

8.8 

32.7 

9.4 

32.5 

9.9 

32.3 

10.5 

32.1 

11.1 

31.9 

11.6 

35 

33.8 

9.1 

33.6 

9.6 

33.5 

10.2 

33.3 

10.8 

33.1 

11.4 

32.9 

12.0 

36 

34.8 

9.3 

34.6 

9.9 

34.4 

10.5 

34.2 

11.1 

34.0 

11.7 

33.8 

12.3 

37 

35.7 

9.6 

35.6 

10.2 

35.4 

10.8 

35.2 

11.4 

35.0 

12.0 

34.8 

12.7 

38 

36.7 

9.8 

36.5 

10.5 

36.3 

11.1 

36.1 

11.7 

35.9 

12.4 

35.7 

13.0 

39 

37.7 

10.1 

37.5 

10.7 

37.3 

11.4 

37.1 

12.1 

36.9 

12.7 

36.6 

13.3 

40 

38.6 

10.4 

38.5 

11.0 

38.3 

11.7 

38.0 

12.4 

37.8 

13.0 

37.6 

13.7 

41 

39.6 

10.6 

39.4 

11.3 

39.2 

12.0 

39.0 

12.7 

38.8 

13.3 

38.5 

14.0 

42 

40.6 

10.9 

40.4 

11.6 

40.2 

12.3 

39.9 

13.0 

39.7 

13.7 

39.5 

14.4 

43 

41  5 

11.1 

41.3 

11.9 

41.1 

12.6 

40.9 

13.3 

40.7 

14.0 

40.4 

147 

44  ' 

42  5 

11.4 

42.3 

12.1 

42.1 

12.9 

41.8 

13.6 

41.6 

14.3 

41.3 

150 

45 

43.5 

11.6 

43.3 

12.4 

43.0 

13.2 

42.8 

13.9 

42.5 

14.7 

42.3 

15.4 

46 

44.4 

11.9 

44.2 

12.7 

44.0 

13.4 

43.7 

14.2 

43.5 

15.0 

43.2 

15.7 

47 

45.4 

12.2 

45.2 

13.0 

44.9 

13.7 

44.7 

14.5 

44.4 

15.3 

44.2 

16.1 

48 

46.4 

12.4 

46.1 

13.2 

45.9 

14.0 

45.7 

14.8 

45.4 

15.6 

45.1 

16.4 

49 

47.3 

12.7 

47.1 

13.5 

46.9 

14.3 

46.6 

15.1 

46.3 

16.0 

46.0 

16.8 

50 

48.3 

12.9 

48.1 

13.8 

47.8 

14.6 

47.6 

15.5 

47.3 

16.3 

47.0 

17.1 

100 

96.6 

25.9 

96.1 

27.6 

95.6 

29.2 

95.1 

30.9 

94.6 

32.6 

94.0 

34.2 

200 

193.2 

51.8 

192.3 

55.1 

191.3 

58.5 

190.2 

61.8 

189.1 

65.1 

187.9 

68.4 

300 

289.8 

77.6 

288.4 

82.7 

286.9 

87.7 

285.3 

92.7 

283.7 

97.7 

281.9 

102.6 

400 

386.3 

103.5 

384.5 

110.2 

382.5 

117.0 

380.4 

123.6 

378.2 

130.2 

375.9 

136.8 

500 

483.0 

129.4 

480.6 

137.8 

478.1 

146.2 

475.5 

154.5 

472.8 

162.8 

469.9 

171.0 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(105°,  255°, 

(106°,  254°, 

(107°,  253°, 

(108°,  252°, 

(109°,  251°, 

(110°,  250°, 

285°) 

286°) 

287°) 

288°) 

289°) 

290°) 

75° 

74° 

6|  Pt.  73° 

72° 

71° 

6J  Pt.  70° 

Table  1.    Traverse  Table 


159 


15° 

16° 

1§  Pt.  17° 

18° 

19° 

l|Pt.20° 

(165°,  195°, 

(164°,  196°, 

(163°,  197°, 

(162°,  198°, 

(161°,  199°, 

(160°,  200°, 

DlST. 

345°) 

344°) 

343°) 

342°) 

341°) 

340°) 

Lat. 

Dep. 

Lat. 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

49.3 

13.2 

49.0 

14.1 

48.8 

14.9 

48.5 

15.8 

48.2 

16.6 

47.9 

17.4 

52 

50.2 

13.5 

50.0 

14.3 

49.7 

15.2 

49.5 

16.1 

49.2 

16.9 

48.9 

17.8 

53 

51.2 

13.7 

50.9 

14.6 

50.7 

15.5 

50.4 

16.4 

50.1 

17.3 

49.8 

18.1 

54 

52.2 

14.0 

51.9 

14.9 

51.6 

15.8 

51.4 

16.7 

51.1 

17.6 

50.7 

18.5 

55 

53.1 

14.2 

52.9 

15.2 

52.6 

16.1 

52.3 

17.0 

52.0 

17.9 

51.7 

18.8 

56 

54.1 

14.5 

53.8 

15.4 

53.6 

16.4 

53.3 

17.3 

52.9 

18.2 

52.6 

19.2 

57 

55.1 

14.8 

54.8 

15.7 

54.5 

16.7 

54.2 

17.6 

53.9 

18.6 

53.6 

19.5 

58 

56.0 

15.0 

55.8 

16.0 

55.5 

17.0 

55.2 

17.9 

54.8 

18.9 

54.5 

19.8 

59 

57.0 

15.3 

56.7 

16.3 

56.4 

17.2 

56.1 

18.2 

55.8 

19.2 

55.4 

20.2 

60 

58.0 

15.5 

57.7 

16.5 

57.4 

17.5 

57.1 

18.5 

56.7 

19.5 

56.4 

20.5 

61 

58.9 

15.8 

58.6 

16.8 

58.3 

17.8 

58.0 

18.9 

57.7 

19.9 

57.3 

20.9 

62 

59.9 

16.0 

59.6 

17.1 

59.3 

18.1 

59.0 

19.2 

5S.6 

20.2 

58.3 

21.2 

63 

60.9 

16.3 

60.6 

17.4 

60.2 

18.4 

59.9 

19.5 

59.6 

20.5 

59.2 

21.5 

64 

61.8 

16.6 

61.5 

17.6 

61.2 

18.7 

60.9 

19.8 

60.5 

20.8 

60.1 

21.9 

65 

62.8 

16.8 

62.5 

17.9 

62.2 

19.0 

61.8 

20.1 

61.5 

21.2 

61.1 

22.2 

66 

63.8 

17.1 

63.4 

18.2 

63.1 

19.3 

62.8 

20.4 

62.4 

21.5 

62.0 

22.6 

67 

64.7 

17.3 

64.4 

18.5 

64.1 

19.6 

63.7 

20.7 

63.3 

21.8 

63.0 

22.9 

68 

65.7 

17.6 

65.4 

18.7 

65.0 

19.9 

64.7 

21.0 

64.3 

22.1 

63.9 

23.3 

69 

66.6 

17.9 

66.3 

19.0 

66.0 

20.2 

65.6 

21.3 

65.2 

22.5 

64.8 

23.6 

70 

67.6 

18.1 

67.3 

19.3 

66.9 

20.5 

66.6 

21.6 

66.2 

22.8 

65.8 

23.9 

71 

68.6 

18.4 

68.2 

19.6 

67.9 

20.8 

67.5 

21.9 

67.1 

23.1 

66.7 

24.3 

72 

69.5 

18.6 

69.2 

19.8 

68.9 

21.1 

68.5 

22.2 

68.1 

23.4 

67.7 

24.6 

73 

70.5 

18.9 

70.2 

20.1 

69.8 

21.3 

69.4 

22.6 

69.0 

23.8 

68.6 

25.0 

74 

71.5 

19.2 

71.1 

20.4 

70.8 

21.6 

70.4 

22.9 

70.0 

24.1 

69.5 

25.3 

75 

72.4 

19.4 

72.1 

20.7 

71.7 

21.9 

71.3 

23.2 

70.9 

24.4 

70.5 

25.7 

76 

73.4 

19.7 

73.1 

20.9 

72.7 

22.2 

72.3 

23.5 

71.9 

24.7 

71.4 

26.0 

77 

74.4 

19.9 

74.0 

21.2 

73.6 

22.5 

73.2 

23.8 

72.8 

25.1 

72.4 

26.3 

78 

75.3 

20.2 

75.0 

21.5 

74.6 

22.8 

74.2 

24.1 

73.8 

25.4 

73.3 

26.7 

79 

76.3 

20.4 

75.9 

21.8 

75.5 

23.1 

75.1 

24.4 

74.7 

25.7 

74.2 

27.0 

80 

77.3 

20.7 

76.9 

22.1 

76.5 

23.4 

76.1 

24.7 

75.6 

26.0 

75.2 

27.4 

81 

78.2 

21.0 

77.9 

22.3 

77.5 

23.7 

77.0 

25.0 

76.6 

26.4 

76.1 

27.7 

82 

79.2 

21.2 

78.8 

22.6 

78.4 

24.0 

78.0 

25.3 

77.5 

26.7 

77.1 

28.0 

83 

80.2 

21.5 

79.8 

22.9 

79.4 

24.3 

78.9 

25.6 

78.5 

27.0 

78.0 

28.4 

84 

81.1 

21.7 

80.7 

23.2 

80.3 

24.6 

79.9 

26.0 

79.4 

27.3 

78.9 

28.7 

85 

82.1 

22.0 

81.7 

23.4 

81.3 

24.9 

80.8 

26.3 

80.4 

27.7 

79.9 

29.1 

86 

83.1 

22.3 

82.7 

23.7 

82.2 

25.1 

81.8 

26.6 

81.3 

28.0 

80.8 

29.4 

87 

84.0 

22.5 

83.6 

24.0 

83.2 

25.4 

82.7 

26.9 

82.3 

28.3 

81.8 

29.8 

88 

85.0 

22.8 

84.6 

24.3 

84.2 

25.7 

83.7 

27.2 

83.2 

28.7 

82.7 

30.1 

80 

86.0 

23.0 

85.6 

24.5 

85.1 

26.0 

84.6 

27.5 

84.2 

29.0 

83.6 

30.4 

90 

86.9 

23.3 

86.5 

24.8 

86.1 

26.3 

85.6 

27.8 

85.1 

29.3 

84.6 

30.8 

01 

87.9 

23.6 

87.5 

25.1 

87.0 

26.6 

86.5 

28.1 

86.0 

29.6 

85.5 

31.1 

92 

88.9 

23.8 

88.4 

25.4 

88.0 

26.9 

87.5 

28.4 

87.0 

30.0 

86.5 

31.5 

93 

89.8 

24.1 

89.4 

25.6 

88.9 

27.2 

88.4 

28.7 

87.9 

30.3 

87.4 

31.8 

94 

90.8 

24.3 

90.4 

25.9 

89.9 

27.5 

89.4 

29.0 

88.9 

30.6 

88.3 

32.1 

95 

91.8 

24.6 

91.3 

26.2 

90.8 

27.8 

90.4 

29.4 

89.8 

30.9 

89.3 

32.5 

96 

92.7 

24.8 

92.3 

26.5 

91.8 

28.1 

91.3 

29.7 

90.8 

31.3 

90.2 

32.8 

97 

93.7 

25.1 

93.2 

26.7 

92.8 

28.4 

92.3 

30.0 

91.7 

31.6 

91.2 

33.2 

98 

94.7 

25.4 

94.2 

27.0 

93.7 

28.7 

93.2 

30.3 

92.7 

31.9 

92.1 

33.5 

99 

95.6 

25.6 

95.2 

27.3 

94.7 

28.9 

94.2 

30.6 

93.6 

32.2 

93.0 

33.9 

100 

96.6 

25.9 

96.1 

27.6 

95.6 

29.2 

95.1 

30.9 

94.6 

32.6 

94.0 

34.2 

600 

579.5 

155.3 

576.8 

165.4 

573.8 

175.4 

570.6 

185.4 

567.3 

195.3 

563.8 

205.2 

700 

676.1 

181.1 

672.8 

193.0 

669.4 

204.6 

665.8 

216.3 

661.9 

227.9 

657.9 

239.4 

800 

772.7 

207.0 

769.0 

220.5 

765.0 

233.9 

760.8 

247.3 

756.5 

260.4 

751.8 

273.6 

900 

869.2 

232.9 

865.0 

248.0 

860.6 

263.1 

855.9 

278.1 

850.9 

292.9 

845.7 

307.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(105°,  255°, 

(106°,  254°, 

(107°,  253°, 

(108°,  252°, 

(109°,  251°, 

(110°,  250°, 

285°) 

286°) 

287°) 

288°) 

289°) 

290°) 

75° 

74° 

6|  Pt,  73° 

72° 

71° 

70° 

160 


Table  1.    Traverse  Table 


21° 

22° 

2  Pt.  23° 

24° 

2|Pt.25° 

26° 

DlST. 

(159°,  201°, 

(158°,  202°, 

(157°,  203°, 

(156°,  204°, 

(155°,  205°, 

(154°,  206°, 

339°) 

338°) 

337°) 

336°) 

335°) 

334°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.9 

0.4 

0.9 

0.4 

0.9 

0.4 

0.9 

0.4 

0.9 

0.4 

0.9 

0.4 

2 

1.9 

0.7 

1.9 

0.7 

1.8 

0.8 

1.8 

0.8 

1.8 

0.8 

1.8 

0.9 

3 

2.8 

1.1 

2.8 

1.1 

2.8 

1.2 

2.7 

1.2 

2.7 

1.3 

2.7 

1.3 

4 

3.7 

1.4 

3.7 

1.5 

3.7 

1.6 

3.7 

1.6 

3.6 

1.7 

3.6 

1.8 

5 

4.7 

1.8 

4.6 

1.9 

4.6 

2.0 

4.6 

2.0 

4.5 

2.1 

4.5 

2.2 

6 

5.6 

2.2 

5.6 

2.2 

5.5 

2.3 

5.5 

2.4 

5.4 

2.5 

5.4 

2.6 

7 

6.5 

2.5 

6.5 

2.6 

6.4 

2.7 

6.4 

2.8 

6.3 

3.0 

6.3 

3.1 

8 

7.5 

2.9 

7.4 

3.0 

7.4 

3.1 

7.3 

3.3 

7.3 

3.4 

7.2 

3.5 

9 

8.4 

3.2 

8.3 

3.4 

8.3 

3.5 

8.2 

3.7 

8.2 

3.8 

8.1 

3.9 

10 

9.3 

3.6 

9.3 

3.7 

9.2 

3.9 

9.1 

4.1 

9.1 

4.2 

9.0 

4.4 

11 

10.3 

3.9 

10.2 

4.1 

10.1 

4.3 

10.0 

4.5 

10.0 

4.6 

9.9 

4.8 

12 

11.2 

4.3 

11.1 

4.5 

11.0 

4.7 

11.0 

4.9 

10.9 

5.1 

10.8 

5.3 

13 

12.1 

4.7 

12.1 

4.9 

12.0 

5.1 

11.9 

5.3 

11.8 

5.5 

11.7 

5.7 

14 

13.1 

5.0 

13.0 

5.2 

12.9 

5.5 

12.8 

5.7 

12.7 

5.9 

12.6 

6.1 

15 

14.0 

5.4 

13.9 

5.6 

13.8 

5.9 

13.7 

6.1 

13.6 

6.3 

13.5 

6.6 

16 

14.9 

5.7 

14.8 

6.0 

14.7 

6.3 

14.6 

6.5 

14.5 

6.8 

14.4 

7.0 

17 

15.9 

6.1 

15.8 

6.4 

15.6 

6.6 

15.5 

6.9 

15.4 

7.2 

15.3 

7.5 

18 

16.8 

6.5 

16.7 

6.7 

16.6 

7.0 

16.4 

7.3 

16.3 

7.6 

16.2 

7.9 

19 

17.7 

6.8 

17.6 

7.1 

17.5 

7.4 

17.4 

7.7 

17.2 

8.0 

17.1 

8.3 

20 

18.7 

7.2 

18.5 

7.5 

18.4 

7.8 

18.3 

8.1 

18.1 

8.5 

18.0 

8.8 

21 

19.6 

7.5 

19.5 

7.9 

19.3 

8.2 

19.2 

8.5 

19.0 

8.9 

18.9 

9.2 

22 

20.5 

7.9 

20.4 

8.2 

20.3 

8.6 

20.1 

8.9 

19.9 

9.3 

19.8 

9.6 

23 

21.5 

8.2 

21.3 

8.6 

21.2 

9.0 

21.0 

9.4 

20.8 

9.7 

20.7 

10.1 

24 

22.4 

8.6 

22.3 

9.0 

22.1 

9.4 

21.9 

9.8 

21.8 

10.1 

21.6 

10.5 

25 

23.3 

9.0 

23.2 

9.4 

23.0 

9.8 

22.8 

10.2 

22.7 

10.6 

22.5 

11.0 

26 

24.3 

9.3 

24.1 

9.7 

23.9 

10.2 

23.8 

10.6 

23.6 

11.0 

23.4 

11.4 

27 

25.2 

9.7 

25.0 

10.1 

24.9 

10.5 

24.7 

11.0 

24.5 

11.4 

24.3 

11.8 

28 

26.1 

10.0 

26.0 

10.5 

25.8 

10.9 

25.6 

11.4 

25.4 

11.8 

25.2 

12.3 

29 

27.1 

10.4 

26.9 

10.9 

26.7 

11.3 

26.5 

11.8 

26.3 

12.3 

26.1 

12.7 

30 

28.0 

10.8 

27.8 

11.2 

27.6 

11.7 

27.4 

12.2 

27.2 

12.7 

27.0 

13.2 

31 

28.9 

11.1 

28.7 

11.6 

28.5 

12.1 

28.3 

12.6 

28.1 

13.1 

27.9 

13.6 

32 

29.9 

11.5 

29.7 

12.0 

29.5 

12.5 

29.2 

13.0 

29.0 

13.5 

28.8 

14.0 

33 

30.8 

11.8 

30.6 

12.4 

30.4 

12.9 

3o!i 

13.4 

29.9 

13.9 

29.7 

14.5 

34 

31.7 

12.2 

31.5 

12.7 

31.3 

13.3 

31.1 

13.8 

30.8 

14.4 

30.6 

14.9 

35 

32.7 

12.5 

32.5 

13.1 

32.2 

13.7 

32.0 

14.2 

31.7 

14.8 

31.5 

15.3 

36 

33.6 

12.9 

33.4 

13.5 

33.1 

14.1 

32.9 

14.6 

32.6 

15.2 

32.4 

15.8 

37 

34.5 

13.3 

34.3 

13.9 

34.1 

14.5 

33.8 

15.0 

33.5 

15.6 

33.3 

16.2 

38 

35.5 

13.6 

35.2 

14.2 

35.0 

14.8 

34.7 

15.5 

34.4 

16.1 

34.2 

16.7 

39 

36.4 

14.0 

36.2 

14.6 

35.9 

15.2 

35.6 

15.9 

35.3 

16.5 

35.1 

17.1 

40 

37.3 

14.3 

37.1 

15.0 

36.8 

15.6 

36.5 

16.3 

36.3 

16.9 

36.0 

17.5 

41 

38.3 

14.7 

38.0 

15.4 

37.7 

16.0 

37.5 

16.7 

37.2 

17.3 

36.9 

18.0 

42 

39.2 

15.1 

38.9 

15.7 

38.7 

16.4 

38.4 

17.1 

38.1 

17.7 

37.7 

18.4 

43 

40.1 

15.4 

39.9 

16.1 

39.6 

16.8 

39.3 

17.5 

39.0 

18.2 

38.6 

18.8 

44 

41.1 

15.8 

40.8 

16.5 

40.5 

17.2 

40.2 

17.9 

39.9 

18.6 

39.5 

19.3 

45 

42.0 

16.1 

41.7 

16.9 

41.4 

17.6 

41.1 

18.3 

40.8 

19.0 

40.4 

19.7 

46 

42.9 

16.5 

42.7 

17.2 

42.3 

18.0 

42.0 

18.7 

41.7 

19.4 

41.3 

20.2 

47 

43.9 

16.8 

43.6 

17.6 

43.3 

18.4 

42.9 

19.1 

42.6 

19.9 

42.2 

20.6 

48 

44.8 

17.2 

44.5 

18.0 

44.2 

18.8 

43.9 

19.5 

43.5 

20.3 

43.1 

21.0 

49 

45.7 

17.6 

45.4 

18.4 

45.1 

19.1 

44.8 

19.9 

44.4 

20.7 

44.0 

21.5 

50 

46.7 

17.9 

46.4 

18.7 

46.0 

19.5 

45.7 

20.3 

45.3 

21.1 

44.9 

21.9 

100 

93.4 

35.8 

92.7 

37.5 

92.1 

39.1 

91.4 

40.7 

90.6 

42.3 

89.9 

43.8 

200 

186.7 

71.7 

185.4 

74.9 

184.1 

78.1 

182.7 

81.3 

181.3 

84.5 

179.8 

87.7 

300 

280.1 

107.5 

278.2 

112.4 

276.2 

117.2 

274.1 

122.0 

271.9 

126.8 

269.6 

131.5 

400 

373.4 

143.4 

370.9 

149.8 

368.2 

156.3 

365.4 

162.7 

362.5 

169.0 

359.5 

175.4 

500 

466.8 

179.2 

463.6 

187.3 

460.2 

195.4 

456.8 

203.4 

453.1 

211.3 

449.4 

219.2 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(111°,  249°, 

(112°,  248,° 

(113°.  247°, 

(114°,  246°, 

(115°,  245°, 

(116°,  244°, 

291°) 

292°) 

293°) 

294°) 

295°) 

296°) 

69° 

6  Pt.  68° 

67° 

66° 

5f  Pt.  65° 

64° 

The  2-Pt.  or  23°  Courses  are  :  N.N.E.,  N.N.W.,  S.S.E.,  S.S.W. 


Table  1.    Traverse  Table 


161 


21° 

22° 

2  Pt.  23° 

24° 

2iPt.  25° 

26° 

(159°,  201°, 

(158°,  202°, 

(157°,  203°, 

(156°,  204°, 

(155°,  205°, 

(154°,  206°, 

DlST. 

339°) 

338°) 

337°) 

336°) 

335°) 

334°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

47.6 

18.3 

47.3 

19.1 

46.9 

19.9 

46.6 

20.7 

46.2 

21.6 

45.8 

22.4 

52 

48.5 

18.6 

48.2 

19.5 

47.9 

20.3 

47.5 

21.2 

47.1 

22.0 

46.7 

22.8 

53 

49.5 

19.0 

49.1 

19.9 

48.8 

20.7 

48.4 

21.6 

48.0 

22.4 

47.6 

23.2 

54 

50.4 

19.4 

50.1 

20.2 

49.7 

21.1 

49.3 

22.0 

48.9 

22.8 

48.5 

23.7 

55 

51.3 

19.7 

51.0 

20.6 

50.6 

21.5 

50.2 

22.4 

49.8 

23.2 

49.4 

24.1 

56 

52.3 

20.1 

51.9 

21.0 

51.5 

21.9 

51.2 

22.8 

50.8 

23.7 

50.3 

24.5 

57 

53.2 

20.4 

52.8 

21.4 

52.5 

22.3 

52.1 

23.2 

51.7 

24.1 

51.2 

25.0 

58 

54.1 

20.8 

53.8 

21.7 

53.4 

22.7 

53.0 

23.6 

52.6 

24.5 

52.1 

25.4 

59 

55.1 

21.1 

54.7 

22.1 

54.3 

23.1 

53.9 

24.0 

53.5 

24.9 

53.0 

25.9 

60 

56.0 

21.5 

55.6 

22.5 

55.2 

23.4 

54.8 

24.4 

54.4 

25.4 

53.9 

26.3 

61 

56.9 

21.9 

56.6 

22.9 

56.2 

23.8 

55.7 

24.8 

55.3 

25.8 

54.8 

26.7 

62 

57.9 

22.2 

57.5 

23.2 

57.1 

24.2 

56.6 

25.2 

56.2 

26.2 

55.7 

27.2 

63 

58.8 

22.6 

58.4 

23.6 

58.0 

24.6 

57.6 

25.6 

57.1 

26.6 

56.6 

27.6 

64 

59.7 

22.9 

59.3 

24.0 

58.9 

25.0 

58.5 

26.0 

58.0 

27.0 

57.5 

28.1 

65 

60.7 

23.3 

60.3 

24.3 

59.8 

25.4 

59.4 

26.4 

58.9 

27.5 

58.4 

28.5 

66 

61.6 

23.7 

61.2 

24.7 

60.8 

25.8 

60.3 

26.8 

59.8 

27.9 

59.3 

28.9 

67 

62.5 

24.0 

62.1 

25.1 

61.7 

26.2 

61.2 

27.3 

60.7 

28.3 

60.2 

29.4 

68 

63.5 

24.4 

63.0 

25.5 

62.6 

26.6 

62.1 

27.7 

61.6 

28.7 

61.1 

29.8 

69 

64.4 

24.7 

64.0 

25.8 

63.5 

27.0 

63.0 

28.1 

62.5 

29.2 

62.0 

30.2 

70 

65.4 

25.1 

64.9 

26.2 

64.4 

27.4 

63.9 

28.5 

63.4 

29.6 

62.9 

30.7 

71 

66.3 

25.4 

65.8 

26.6 

65.4 

27.7 

64.9 

28.9 

64.3 

30.0 

63.8 

31.1 

72 

67.2 

25.8 

66.8 

27.0 

66.3 

28.1 

65.8 

29.3 

65.3 

30.4 

64.7 

31.6 

73 

68.2 

26.2 

67.7 

27.3 

67.2 

28.5 

66.7 

29.7 

66.2 

30.9 

65.6 

32.0 

74 

69.1 

26.5 

68.6 

27.7 

68.1 

28.9 

67.6 

30.1 

67.1 

31.3 

66.5 

32.4 

75 

70.0 

26.9 

69.5 

28.1 

69.0 

29.3 

68.5 

30.5 

68.0 

31.7 

67.4 

32.9 

76 

71.0 

27.2 

70.5 

28.5 

70.0 

29.7 

69.4 

30.9 

68.9 

32.1 

68.3 

33.3 

77 

71.9 

27.6 

71.4 

28.8 

70.9 

30.1 

70.3 

31.3 

69.8 

32.5 

69.2 

33.8 

78 

72.8 

28.0 

72.3 

29.2 

71.8 

30.5 

71.3 

31.7 

70.7 

33.0 

70.1 

34.2 

79 

73.0 

28.3 

73.2 

29.6 

72.7 

30.9 

72.2 

32.1 

71.6 

33.4 

71.0 

34.6 

80 

74.7 

28.7 

74.2 

30.0 

73.6 

31.3 

73.1 

32.5 

72.5 

33.8 

71.9 

35.1 

81 

75.6 

29.0 

75.1 

30.3 

74.6 

31.6 

74.0 

32.9 

73.4 

34.2 

72.8 

35.5 

82 

76.6 

29.4 

76.0 

30.7 

75.5 

32.0 

74.9 

33.4 

74.3 

34.7 

73.7 

35.9 

83 

77.5 

29.7 

77.0 

31.1 

76.4 

32.4 

75.8 

33.8 

75.2 

35.1 

74.6 

36.4 

84 

78.4 

30.1 

77.9 

31.5 

77.3 

32.8 

76.7 

34.2 

76.1 

35.5 

75.5 

36.8 

85 

79.4 

30.5 

78.8 

31.8 

78.2 

33.2 

77.7 

34.6 

77.0 

35.9 

76.4 

37.3 

86 

80.3 

30.8 

79.7 

32.2 

79.2 

33.6 

78.6 

35.0 

77.9 

36.3 

77.3 

37.7 

87 

81.2 

31.2 

80.7 

32.6 

80.1 

34.0 

79.5 

35.4 

78.8 

36.8 

78.2 

38.1 

88 

82.2 

31.5 

81.6 

33.0 

81.0 

34.4 

80.4 

35.8 

79.8 

37.2 

79.1 

38.6 

89 

83.1 

31.9 

82.5 

33.3 

81.9 

34.8 

81.3 

36.2 

80.7 

37.6 

80.0 

39.0 

90 

84.0 

32.3 

83.4 

33.7 

82.8 

35.2 

82.2 

36.6 

81.6 

38.0 

80.9 

39.5 

91 

85.0 

32.6 

84.4 

34.1 

83.8 

35.6 

83.1 

37.0 

82.5 

38.5 

81.8 

39.9 

92 

85.9 

33.0 

85.3 

34.5 

84.7 

35.9 

84.0 

37.4 

83.4 

38.9 

82.7 

40.3 

93 

86.8 

33.3 

86.2 

34.8 

85.6 

36.3 

85.0 

37.8 

84.3 

39.3 

83.6 

40.8 

94 

87.8 

33.7 

87.2 

35.2 

86.5 

36.7 

85.9 

38.2 

85.2 

39.7 

84.5 

41.2 

95 

88.7 

34.0 

88.1 

35.6 

87.4 

37.1 

86.8 

38.6 

86.1 

40.1 

85.4 

41.6 

96 

89.6 

34.4 

89.0 

36.0 

88.4 

37.5 

87.7 

39.0 

87.0 

40.6 

86.3 

42.1 

97 

90.6 

34.8 

89.9 

36.3 

89.3 

37.9 

88.6 

39.5 

87.9 

41.0 

87.2 

42.5 

98 

91.5 

35.1 

90.9 

36.7 

90.2 

38.3 

89.5 

39.9 

88.8 

41.4 

88.1 

43.0 

99 

92.4 

35.5 

91.8 

37.1 

91.1 

38.7 

90.4 

40.3 

89.7 

41.8 

89.0 

43.4 

100 

93.4 

35.8 

92.7 

37.5 

92.1 

39.1 

91.4 

40.7 

90.6 

42.3 

89.9 

43.8 

600 

560.1 

215.0 

556.3 

224.8 

552.3 

234.4 

548.1 

244.0 

543.8 

253.6 

539.3 

263.0 

700 

653.6 

250.8 

649.1 

262.2 

644.3 

273.5 

639.5 

284.7 

634.5 

295.8 

629.2 

306.8 

800 

746.9 

286.7 

741.8 

299.7 

736.4 

312.6 

730.8 

325.4 

725.1 

338.1 

719.1 

350.6 

900 

840.3 

322.5 

834.5 

IK  7.1 

828.3 

351.7 

822.1 

366.0 

815.6 

380.3 

808.9 

394.5 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(111°,  249°, 

(112°,  248°, 

(113°,  247°, 

(114°,  246°, 

(115°,  245°, 

(116°,  244°, 

291°) 

292°) 

293°) 

294°) 

295°) 

296°) 

69° 

6  Pt.  68° 

67° 

66° 

5f  Pt.  65° 

64° 

The  6-Pt.  or  68°  Courses  are:  E.N.E.,  W.N.W.,  E.S.E.,  W.S.W. 


162 


Table  1.    Traverse  Table 


27° 

1\  Pt.  28° 

29° 

30° 

2f  Pt.  31° 

32° 

DlST. 

(153°,  207°, 
333°) 

(152°,  208°, 
332°) 

(151°,  209°, 
331°) 

(150°,  210°, 
330°) 

<149°,  211°, 
329°) 

(148°,  212°, 
328°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.8 

0.5 

2 

1.8 

0.9 

1.8 

0.9 

1.7 

1.0 

1.7 

1.0 

1.7 

1.0 

1.7 

1.1 

3 

2.7 

1.4 

2.6 

1.4 

2.6 

1.5 

2.6 

1.5 

2.6 

1.5 

2.5 

1.6 

4 

3.6 

1.8 

3.5 

1.9 

3.5 

1.9 

3.5 

2.0 

3.4 

2.1 

3.4 

2.1 

5 

4.5 

2.3 

4.4 

2.3 

4.4 

2.4 

4.3 

2.5 

4.3 

2.6 

4.2 

2.6 

6 

5.3 

2.7 

5.3 

2.8 

5.2 

2.9 

5.2 

3.0 

5.1 

3.1 

5.1 

3.2 

7 

6.2 

3.2 

6.2 

3.3 

6.1 

3.4 

6.1 

3.5 

6.0 

3.6 

5.9 

3.7 

8 

7.1 

3.6 

7.1 

3.8 

7.0 

3.9 

6.9 

4.0 

6.9 

4.1 

6.8 

4.2 

9 

8.0 

4.1 

7.9 

4.2 

7.9 

4.4 

7.8 

4.5 

Y.7 

4.6 

7.6 

4.8 

10 

8.9 

4.5 

8.8 

4.7 

8.7 

4.8 

8.7 

5.0 

8.6 

5.2 

8.5 

5.3 

11 

9.8 

5.0 

9.7 

5.2 

9.6 

5.3 

9.5 

5.5 

9.4 

5.7 

9.3 

5.8 

12 

10.7 

5.4 

10.6 

5.6 

10.5 

5.8 

10.4 

6.0 

10.3 

6.2 

10.2 

6.4 

13 

11.6 

5.9 

11.5 

6.1 

11.4 

6.3 

11.3 

6.5 

11.1 

6.7 

11.0 

6.9 

14 

12.5 

6.4 

12.4 

6.6 

12.2 

6.8 

12.1 

7.0 

12.0 

7.2 

11.9 

7.4 

15 

13.4 

6.8 

13.2 

7.0 

13.1 

7.3 

13.0 

7.5 

12.9 

7.7 

12.7 

7.9 

16 

14.3 

7.3 

14.1 

7.5 

14.0 

7.8 

13.9 

8.0 

13.7 

8.2 

13.6 

8.5 

17 

15.1 

7.7 

15.0 

8.0 

14.9 

8.2 

14.7 

8.5 

14.6 

8.8 

14.4 

9.0 

18 

16.0 

8.2 

15.9 

8.5 

15.7 

8.7 

15.6 

9.0 

15.4 

9.3 

15.3 

9.5 

19 

16.9 

8.6 

16.8 

8.9 

16.6 

9.2 

16.5 

9.5 

16.3 

9.8 

16.1 

10.1 

20 

17.8 

9.1 

17.7 

9.4 

17.5 

9.7 

17.3 

10.0 

17.1 

10.3 

17.0 

10.6 

21 

18.7 

9.5 

18.5 

9.9 

18.4 

10.2 

18.2 

10.5 

18.0 

10.8 

17.8 

11.1 

22 

19.6 

10.0 

19.4 

10.3 

19.2 

10.7 

19.1 

11.0 

18.9 

11.3 

18.7 

11.7 

23 

20.5 

10.4 

20.3 

10.8 

20.1 

11.2 

19.9 

11.5 

19.7 

11.8 

19.5 

12.2 

24 

21.4 

10.9 

21.2 

11.3 

21.0 

11.6 

20.8 

12.0 

20.6 

12.4 

20.4 

12.7 

25 

22.3 

11.3 

22.1 

11.7 

21.9 

12.1 

21.7 

12.5 

21.4 

12.9 

21.2 

13.2 

26 

23.2 

11.8 

23.0 

12.2 

22  7 

12.6 

22.5 

13.0 

22.3 

13.4 

22.0 

13.8 

27 

24.1 

12.3 

23.8 

12.7 

23^6 

13.1 

23.4 

13.5 

23.1 

13.9 

22.9 

14.3 

28 

24.9 

12.7 

24.7 

13.1 

24.5 

13.6 

24.2 

14.0 

24.0 

14.4 

23.7 

14.8 

29 

25.8 

13.2 

25.6 

13.6 

25.4 

14.1 

25.1 

14.5 

24.9 

14.9 

24.6 

15.4 

30 

26.7 

13.6 

26.5 

14.1 

26.2 

14.5 

26.0 

15.0 

25.7 

15.5 

25.4 

15.9 

31 

27.6 

14.1 

27.4 

14.6 

27.1 

15.0 

26.8 

15.5 

26.6 

16.0 

26.3 

16.4 

32 

28.5 

14.5 

28.3 

15.0 

28.0 

15.5 

27.7 

16.0 

27.4 

16.5 

27.1 

17.0 

33 

29.4 

15.0 

29.1 

15.5 

28.9 

16.0 

28.6 

16.5 

28.3 

17.0 

28.0 

17.5 

34 

30.3 

15.4 

30.0 

16.0 

29.7 

16.5 

29.4 

17.0 

29.1 

17.5 

28.8 

18.0 

35 

31.2 

15.9 

30.9 

16.4 

30.6 

17.0 

30.3 

17.5 

30.0 

18.0 

29.7 

18.5 

36 

32.1 

16.3 

31.8 

16.9 

31.5 

17.5 

31.2 

18.0 

30.9 

18.5 

30.5 

19.1 

37 

33.0 

16.8 

32.7 

17.4 

32.4 

17.9 

32.0 

18.5 

31.7 

19.1 

31.4 

19.6 

38 

33.9 

17.3 

33.6 

17.8 

33.2 

18.4 

32.9 

19.0 

32.6 

19.6 

32.2 

20.1 

39 

34.7 

17.7 

34.4 

18.3 

34.1 

18.9 

33.8 

19.5 

33.4 

20.1 

33.1 

20.7 

40 

35.6 

18.2 

35.3 

18.8 

35.0 

19.4 

34.6 

20.0 

34.3 

20.6 

33.9 

21.2 

41 

36.5 

18.6 

36.2 

19.2 

35.9 

19.9 

35.5 

20.5 

35.1 

21.1 

34.8 

21.7 

42 

37.4 

19.1 

37.1 

19.7 

36.7 

20.4 

36.4 

21.0 

36.0 

21.6 

35.6 

22.3 

43 

38.3 

19.5 

38.0 

20.2 

37.6 

20.8 

37.2 

21.5 

36.9 

22.1 

36.5 

22.8 

44 

39.2 

20.0 

38.8 

20.7 

38.5 

21.3 

38.1 

22.0 

37.7 

22.7 

37.3 

23.3 

45 

40.1 

20.4 

39.7 

21.1 

39.4 

21.8 

39.0 

22.5 

38.6 

23.2 

38.2 

23.8 

46 

41.0 

20.9 

40.6 

21.6 

40.2 

22.3 

39.8 

23.0 

39.4 

23.7 

39.0 

24.4 

47 

41.9 

21.3 

41.5 

22.1 

41.1 

22.8 

40.7 

23.5 

40.3 

24.2 

39.9 

24.9 

48 

42.8 

21.8 

42.4 

22.5 

42.0 

23.3 

41.6 

24.0 

41.1 

24.7 

40.7 

25.4 

49 

43.7 

22.2 

43.3 

23.0 

42.9 

23.8 

42.4 

24.5 

42.0 

25.2 

41.6 

26.0 

50 

44.6 

22.7 

44.1 

23.5 

43.7 

24.2 

43.3 

25.0 

42.9 

25.8 

42.4 

26.5 

100 

89.1 

45.4 

88.3 

46.9 

87.5 

48.5 

86.6 

50.0 

85.7 

51.5 

84.8 

53.0 

200 

178.2 

90.8 

176.6 

93.9 

174.9 

97.0 

173.2 

100.0 

171.4 

103.0 

169.6 

106.0 

300 

267.3 

136.2 

264.9 

140.8 

262.4 

145.4 

259.8 

150.0 

257.1 

154.5 

254.4 

159.0 

400 

356.4 

181.6 

353.1 

187.8 

349.8 

193.9 

346.4 

200.0 

342.9 

206.0 

339.2 

211.9 

500 

445.5 

227.0 

441.5 

234.7 

437.3 

242.4 

433.0 

250.0 

428.6 

257.5 

424.0 

265.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(117°,  243°, 

(118°,  242°, 

(119°,  241°, 

(120°,  240°, 

(121°,  239°, 

(122°,  238°, 

297°) 

298°) 

299°) 

300°) 

301°) 

302°) 

63° 

5|Pt.  62° 

61° 

60° 

5|Pt.  59° 

58° 

Table  1.    Traverse  Table 


163 


27° 

2£  Pt.  28° 

29° 

30° 

2f  Pt.  31° 

32° 

(153°,  207°, 

(152°,  208°, 

(151°,  209°, 

(150°,  210°, 

(149°,  211°, 

(148°,  212°, 

DlST. 

333°) 

332°) 

331°) 

330°) 

329°) 

328°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

45.4 

23.2 

45.0 

23.9 

44.6 

24.7 

44.2 

25.5 

43.7 

26.3 

43.3 

27.0 

52 

46.3 

23.6 

45.9 

24.4 

45.5 

25.2 

45.0 

26.0 

44.6 

26.8 

44.1 

27.6 

53 

47.2 

24.1 

46.8 

24.9 

46.4 

25.7 

45.9 

26.5 

45.4 

27.3 

44.9 

28.1 

54 

48.1 

24.5 

47.7 

25.4 

47.2 

26.2 

46.8 

27.0 

46.3 

27.8 

45.8 

28.6 

55 

49.0 

25.0 

48.6 

25.8 

48.1 

26.7 

47.6 

27.5 

47.1 

28.3 

46.6 

29.1 

56 

49.9 

25.4 

49.4 

26.3 

49.0 

27.1 

48.5 

28.0 

48.0 

28.8 

47.5 

29.7 

57 

50.8 

25.9 

50.3 

26.8 

49.9 

27.6 

49.4 

28.5 

48.9 

29.4 

48.3 

30.2 

58 

51.7 

26.3 

51.2 

27.2 

50.7 

28.1 

50.2 

29.0 

49.7 

29.9 

49.2 

30.7 

59 

52.6 

26.8 

52.1 

27.7 

51.6 

28.6 

51.1 

29.5 

50.6 

30.4 

50.0 

31.3 

60 

53.5 

27.2 

53.0 

28.2 

52.5 

29.1 

52.0 

30.0 

51.4 

30.9 

50.9 

31.8 

61 

54.4 

27.7 

53.9 

28.6 

53.4 

29.6 

52.8 

30.5 

52.3 

31.4 

51.7 

32.3 

62 

55.2 

28.1 

54.7 

29.1 

54.2 

30.1 

53.7 

31.0 

53.1 

31.9 

52.6 

32.9 

63 

56.1 

28.6 

55.6 

29.6 

55.1 

30.5 

54.6 

31.5 

54.0 

32.4 

53.4 

33.4 

64 

57.0 

29.1 

56.5 

30.0 

56.0 

31.0 

55.4 

32.0 

54.9 

33.0 

54.3 

33.9 

65 

57.9 

29.5 

57.4 

30.5 

56.9 

31.5 

56.3 

32.5 

55.7 

33.5 

55.1 

34.4 

66 

58.8 

30.0 

58.3 

31.0 

57.7 

32.0 

57.2 

"33.0 

56.6 

34.0 

56.0 

35.0 

67 

59.7 

30.4 

59.2 

31.5 

58.6 

32.5 

58.0 

33.5 

57.4 

34.5 

56.8 

35.5 

68 

60.6 

30.9 

60.0 

31.9 

59.5 

33.0 

58.9 

34.0 

58.3 

35.0 

57.7 

36.0 

69 

61.5 

31.3 

60.9 

32.4 

60.3 

33.5 

59.8 

34.5 

59.1 

35.5 

58.5 

36.6 

70 

62.4 

31.8 

61.8 

32.9 

61.2 

33.9 

60.6 

35.0 

60.0 

36.1 

59.4 

37.1 

71 

63.3 

32.2 

62.7 

33.3 

62.1 

34.4 

61.5 

35.5 

60.9 

36.6 

60.2 

37.6 

72 

64.2 

32.7 

63.6 

33.8 

63.0 

34.9 

62.4 

36.0 

61.7 

37.1 

61.1 

38.2 

73 

65.0 

33.1 

64.5 

34.3 

63.8 

35.4 

63.2 

36.5 

62.6 

37.6 

61.9 

38.7 

74 

65.9 

33.6 

65.3 

34.7 

64.7 

35.9 

64.1 

37.0 

63.4 

38.1 

62.8 

39.2 

75 

66.8 

34.0 

66.2 

35.2 

65.6 

36.4 

65.0 

37.5 

64.3 

38.6 

63.6 

39.7 

76 

67.7 

34.5 

67.1 

35.7 

66.5 

36.8 

65.8 

38.0 

65.1 

39.1 

64.5 

40.3 

77 

68.6 

35.0 

68.0 

36.1 

67.3 

37.3 

66.7 

38.5 

66.0 

39.7 

65.3 

40.8 

78 

69.5 

35.4 

68.9 

36.6 

68.2 

37.8 

67.5 

39.0 

66.9 

40.2 

66.1 

41.3 

79 

70.4 

35.9 

69.8 

37.1 

69.1 

38.3 

68.4 

39.5 

67.7 

40.7 

67.0 

41.9 

80 

71.3 

36.3 

70.6 

37.6 

70.0 

38.8 

69.3 

40.0 

68.6 

41.2 

67.8 

42.4 

81 

72.2 

36.8 

71.5 

38.0 

70.8 

39.3 

70.1 

40.5 

69.4 

41.7 

68.7 

42.9 

82 

73.1 

37.2 

72.4 

38.5 

71.7 

39.8 

71.0 

41.0 

70.3 

42.2 

69.5 

43.5 

83 

74.0 

37.7 

73.3 

39.0 

72.6 

40.2 

71.9 

41.5 

71.1 

42.7 

70.4 

44.0 

84 

74.8 

38.1 

74.2 

39.4 

73.5 

40.7 

72.7 

42.0 

72.0 

43.3 

71.2 

44.5 

85 

75.7 

38.6 

75.1 

39.9 

74.3 

41.2 

73.6 

42.5 

72.9 

43.8 

72.1 

45.0 

86 

76.6 

39.0 

75.9 

40.4 

75.2 

41.7 

74.5 

43.0 

73.7 

44.3 

72.9 

45.6 

87 

77.5 

39.5 

76.8 

40.8 

76.1 

42.2 

75.3 

43.5 

74.6 

44.8 

73.8 

46.1 

88 

78.4 

40.0 

77.7 

41.3 

77.0 

42.7 

76.2 

44.0 

75.4 

45.3 

74.6 

46.6 

89 

79.3 

40.4 

78.6 

41.8 

77.8 

43.1 

77.1 

44.5 

76.3 

45.8 

75.5 

47.2 

90 

80.2 

40.9 

79.5 

42.3 

78.7 

43.6 

77.9 

45.0 

77.1 

46.4 

76.3 

47.7 

91 

81.1 

41.3 

80.3 

42.7 

79.6 

44.1 

78.8 

45.5 

78.0 

46.9 

77.2 

48.2 

92 

82.0 

41.8 

81.2 

43.2 

80.5 

44.6 

79.7 

46.0 

78.9 

47.4 

78.0 

48.8 

93 

82.9 

42.2 

82.1 

43.7 

81.3 

45.1 

80.5 

46.5 

79.7 

47.9 

78.9 

49.3 

94 

83.8 

42.7 

83.0 

44.1 

82.2 

45.6 

81.4 

47.0 

80.6 

48.4 

79.7 

49.8 

95 

84.6 

43.1 

83.9 

44.6 

83.1 

46.1 

82.3 

47.5 

81.4 

48.9 

80.6 

50.3 

96 

85.5 

43.6 

84.8 

45.1 

84.0 

46.5 

83.1 

48.0 

82.3 

49.4 

81.4 

50.9 

97 

86.4 

44.0 

85.6 

45.5 

84.8 

47.0 

84.0 

48.5 

83.1 

50.0 

82.3 

51.4 

98 

87.3 

44.5 

86.5 

46.0 

85.7 

47.5 

84.9 

49.0 

84.0 

50.5 

83.1 

51.9 

99 

88.2 

44.9 

87.4 

46.5 

86.6 

48.0 

85.7 

49.5 

84.9 

51.0 

84.0 

52.5 

100 

89.1 

45.4 

88.3 

46.9 

87.5 

48.5 

86.6 

50.0 

85.7 

51.5 

84.8 

53.0 

600 

534.6 

272.4 

529.8 

281.7 

524.8 

290.9 

519.6 

300.0 

514.3 

309.0 

508.8 

318.0 

700 

623.7 

317.8 

618.0 

328.6 

612.2 

339.4 

606.1 

350.0 

600.1 

360.4 

593.6 

371.0 

800 

712.9 

363.2 

706.3 

375.6 

699.7 

387.9 

692.8 

400.0 

685.8 

412.0 

678.4 

423.9 

900 

801.9 

408.5 

794.5 

422.5 

787.0 

436.3 

779.3 

450.0 

771.4 

463.4 

763.2 

476.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

(117°,  243°, 

(118°,  242°, 

(119°,  241°, 

(120°,  240° 

(121°,  239°, 

(122°,  238°, 

297°) 

298°) 

299°) 

300°) 

301°) 

302°) 

63° 

5^  Pt.  62° 

61° 

60° 

5|  Pt.  59° 

58° 

164 


Table  1.    Traverse  Table 


33° 

3  Pt.  34° 

35° 

36° 

31  Pt.  37° 

38° 

(147°,  213°, 

(146°,  214°, 

(145°,  215°, 

(144°,  216°, 

(143°,  217°, 

(142°,  218°, 

DlST. 

327°) 

326°) 

325°) 

324°) 

323°) 

322°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.8 

0.5 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

2 

1.7 

1.1 

1.7 

1.1 

1.6 

1.1 

1.6 

1.2 

1.6 

1.2 

1.6 

1.2 

3 

2.5 

1.6 

2.5 

1.7 

2.5 

1.7 

2.4 

1.8 

2.4 

1.8 

2.4 

1.8 

4 

3.4 

2.2 

3.3 

2.2 

3.3 

2.3 

3.2 

2.4 

3.2 

2.4 

3.2 

2.5 

5 

4.2 

2.7 

4.1 

2.8 

4.1 

2.9 

4.0 

2.9 

4.0 

3.0 

3.9 

3.1 

6 

5.0 

3.3 

5.0 

3.4 

4.9 

3.4 

4.9 

3.5 

4.8 

3.6 

4.7 

3.7 

7 

5.9 

3.8 

5.8 

3.9 

5.7 

4.0 

5.7 

4.1 

5.6 

4.2 

5.5 

4.3 

8 

6.7 

4.4 

6.6 

4.5 

6.6 

4.6 

6.5 

4.7 

6.4 

4.8 

6.3 

4.9 

9 

75 

4.9 

7.5 

5.0 

7.4 

5.2 

7.3 

5.3 

7.2 

5.4 

7  1 

5.5 

10 

8.4 

5.4 

8.3 

5.6 

8.2 

5.7 

8.1 

5.9 

8.0 

6.0 

7.9 

6.2 

11 

9.2 

6.0 

9.1 

6.2 

9.0 

6.3 

8.9 

6.5 

8.8 

6.6 

8.7 

6.8 

12 

10.1 

6.5 

9.9 

6.7 

9.8 

6.9 

9.7 

7.1 

9.6 

7.2 

9.5 

7.4 

13 

10.9 

7.1 

10.8 

7.3 

10.6 

7.5 

10.5 

7.6 

10.4 

7.8 

10.2 

8.0 

14 

11.7 

7.6 

11.6 

7.8 

11.5 

8.0 

11.3 

8.2 

11.2 

8.4 

11.0 

8.6 

15 

12.6 

8.2 

12.4 

8.4 

12.3 

8.6 

12.1 

8.8 

12.0 

9.0 

11.8 

9.2 

16 

134 

8.7 

13.3 

8.9 

13.1 

9.2 

12.9 

9.4 

12.8 

9.6 

12.6 

99 

17 

14.3 

9.3 

14.1 

9.5 

13.9 

9.8 

13.8 

10.0 

13.6 

10.2 

13.4 

10.5 

18 

15.1 

9.8 

14.9 

10.1 

14.7 

10.3 

14.6 

10.6 

14.4 

10.8 

14.2 

11.1 

19 

15.9 

10.3 

15.8 

10.6 

15.6 

10.9 

15.4 

11.2 

15.2 

11.4 

15.0 

11.7 

20 

16.8 

10.9 

16.6 

11.2 

16.4 

11.5 

16.2 

11.8 

16.0 

12.0 

15.8 

12.3 

21 

17.6 

11.4 

17.4 

11.7 

17.2 

12.0 

17.0 

12.3 

16.8 

12.6 

16.5 

12.9 

22 

18.5 

12.0 

18.2 

12.3 

18.0 

12.6 

17.8 

12.9 

17.6 

13.2 

17.3 

13.5 

23 

19.3 

12.5 

19.1 

12.9 

18.8 

13.2 

18.6 

13.5 

18.4 

13.8 

18.1 

14.2 

24 

20.1 

13.1 

19.9 

13.4 

19.7 

13.8 

19.4 

14.1 

19.2 

14.4 

18.9 

14.8 

25 

21.0 

13.6 

20.7 

14.0 

20.5 

14.3 

20.2 

14.7 

20.0 

15.0 

19.7 

15.4 

26 

21.8 

14.2 

21.6 

14.5 

21.3 

14.9 

21.0 

15.3 

20.8 

15.6 

20.5 

16.0 

27 

22.6 

14.7 

22.4 

15.1 

22.1 

15.5 

21.8 

15.9 

21.6 

16.2 

21.3 

16.6 

28 

23.5 

15.2 

23.2 

15.7 

22.9 

16.1 

22.7 

16.5 

22.4 

16.9 

22.1 

17.2 

29 

24.3 

15.8 

24.0 

16.2 

23.8 

16.6 

23.5 

17.0 

23.2 

17.5 

22.9 

17.9 

30 

25.2 

16.3 

24.9 

16.8 

24.6 

17.2 

24.3 

17.6 

24.0 

18.1 

23.6 

18.5 

31 

26.0 

16.9 

25.7 

17.3 

25.4 

17.8 

25.1 

18.2 

24.8 

18.7 

24.4 

19.1 

32 

26.8 

17.4 

26.5 

17.9 

26.2 

18.4 

25.9 

18.8 

25.6 

19.3 

25.2 

19.7 

33 

27.7 

18.0 

27.4 

18.5 

27.0 

18.9 

26.7 

19.4 

26.4 

19.9 

26.0 

20.3 

34 

28.5 

18.5 

28.2 

19.0 

27.9 

19.5 

27.5 

20.0 

27.2 

20.5 

26.8 

20.9 

35 

29.4 

19.1 

29.0 

19.6 

28.7 

20.1 

28.3 

20.6 

28.0 

21.1 

27.6 

21.5 

36 

30.2 

19.6 

29.8 

20.1 

29.5 

20.6 

29.1 

21.2 

28.8 

21.7 

28.4 

22.2 

37 

31.0 

20.2 

30.7 

20.7 

30.3 

21.2 

29.9 

21.7 

29.5 

22.3 

29.2 

22.8 

38 

31.9 

20.7 

31.5 

21.2 

31.1 

21.8 

30.7 

22.3 

30.3 

22.9 

29.9 

23.4 

39 

32.7 

21.2 

32.3 

21.8 

31.9 

22.4 

31.6 

22.9 

31.1 

23.5 

30.7 

24.0 

40 

33.5 

21.8 

33.2 

22.4 

32.8 

22.9 

32.4 

23.5 

31.9 

24.1 

31.5 

24.6 

41 

34.4 

22.3 

34.0 

22.9 

33.6 

23.5 

33.2 

24.1 

32.7 

24.7 

32.3 

25.2 

42 

35.2 

22.9 

34.8 

23.5 

34.4 

24.1 

34.0 

24.7 

33.5 

25.3 

33.1 

25.9 

43 

36.1 

23.4 

35.6 

24.0 

35.2 

24.7 

34.8 

25.3 

34.3 

25.9 

33.9 

26.5 

44 

36.9 

24.0 

36.5 

24.6 

36.0 

25.2 

35.6 

25.9 

35.1 

26.5 

34.7 

27.1 

45 

37.7 

24.5 

37.3 

25.2 

36.9 

25.8 

36.4 

26.5 

35.9 

27.1 

35.5 

27.7 

46 

38.6 

25.1 

38.1 

25.7 

37.7 

26.4 

37.2 

27.0 

36.7 

27.7 

36.2 

28.3 

47 

39.4 

25.6 

39.0 

26.3 

38.5 

27.0 

38.0 

27.6 

37.5 

28.3 

37.0 

28.9 

48 

40.3 

26.1 

39.8 

26.8 

39.3 

27.5 

38.8 

28.2 

38.3 

28.9 

37.8 

29.6 

49 

41.1 

26.7 

40.6 

27.4 

40.1 

28.1 

39.6 

28.8 

39.1 

29.5 

38.6 

30.2 

50 

41.9 

27.2 

41.5 

28.0 

41.0 

28.7 

40.5 

29.4 

39.9 

30.1 

39.4 

30.8 

100 

83.9 

54.5 

82.9 

55.9 

81.9 

57.4 

80.9 

58.8 

79.9 

60.2 

78.8 

61.6 

200 

167.7 

108.9 

165.8 

111.8 

163.8 

114.7 

161.8 

117.6 

159.7 

120.4 

157.6 

123.1 

300 

251.6 

163.4 

248.7 

167.8 

245.7 

172.1 

242.7 

176.3 

239.6 

180.5 

236.4 

184.7 

400 

335.5 

217.8 

331.6 

223.7 

327.7 

229.4 

323.6 

235.1 

319.4 

240.7 

315.2 

246.3 

500 

419.3 

272.3 

414.5 

279.6 

409.6 

286.8 

404.5 

293.9 

399.3 

300.9 

394.0 

307.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(123°,  237°, 

(124°,  236°, 

(125°,  235°, 

(126°,  234°, 

(127°,  233°, 

(128°,  232°, 

303°) 

304°) 

305°) 

306°) 

307°) 

308°) 

57° 

5  Pt.  56° 

55° 

54° 

4f  Pt,  53° 

52° 

The  3-Pt.  or  34°  Courses  are :  N.E.  by  N.,  N.W.  by  N.,  S.E.  by  S.,  S.W.  by  S. 


Table  1.    Traverse  Table 


165 


33° 

3  Pt,  34° 

35° 

36° 

3iPt.37° 

38° 

DlST. 

(147°,  213°, 
327°) 

!(146°,  214°, 
326°) 

(145°,  215°, 
325°) 

(144°,  216°, 
324°) 

(143°,  217°, 
323°) 

(142°,  218°, 
322°) 

Lat. 

Dep. 

iLat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

42.8 

27.8 

42.3 

28.5 

41.8 

29.3 

41.3 

30.0 

40.7 

30.7 

40.2 

31.4 

52 

43.6 

28.3 

43.1 

29.1 

42.6 

29.8 

42.1 

30.6 

41.5 

31.3 

41.0 

32.0 

53 

44.4 

28.9 

43.9 

29.6 

43.4 

30.4 

42.9 

31.2 

42.3 

31.9 

41.8 

32.6 

54 

45.3 

29.4 

44.8 

30.2 

44.2 

31.0 

43.7 

31.7 

43.1 

32.5 

42.6 

33.2 

55 

46.1 

30.0 

45.6 

30.8 

45.1 

31.5 

44.5 

32.3 

43.9 

33.1 

43.3 

33.9 

56 

47.0 

30.5 

46.4 

31.3 

45.9 

32.1 

45.3 

32.9 

44.7 

33.7 

44.1 

34.5 

57 

47.8 

31.0 

47.3 

31.9 

46.7 

32.7 

46.1 

33.5 

45.5 

34.3 

44.9 

35.1 

58 

48.6 

31.6 

48.1 

32.4 

47.5 

33.3 

46.9 

34.1 

46.3 

34.9 

45.7 

35.7 

59 

49.5 

32.1 

48.9 

33.0 

48.3 

33.8 

47.7 

34.7 

47.1 

35.5 

46.5 

36.3 

60 

50.3 

32.7 

49.7 

33.6 

49.1 

34.4 

48.5 

35.3 

47.9 

36.1 

47.3 

36.9 

61 

51.2 

33.2 

50.6 

34.1 

50.0 

35.0 

49.4 

35.9 

48.7 

36.7 

48.1 

37.6 

62 

52.0 

33.8 

51.4 

34.7 

50.8 

35.6 

50.2 

36.4 

49.5 

37.3 

48.9 

38.2 

63 

52.8 

34.3 

52.2 

35.2 

51.6 

36.1 

51.0 

37.0 

50.3 

37.9 

49.6 

38.8 

64 

53.7 

34.9 

53.1 

35.8 

52.4 

36.7 

51.8 

37.6 

51.1 

38.5 

50.4 

39.4 

65 

54.5 

35.4 

53.9 

36.3 

53.2 

37.3 

52.6 

38.2 

51.9 

39.1 

51.2 

40.0 

66 

55.4 

35.9 

54.7 

36.9 

54.1 

37.9 

53.4 

38.8 

52.7 

39.7 

52.0 

40.6 

67 

56.2 

36.5 

55.5 

37.5 

54.9 

38.4 

54.2 

39.4 

53.5 

40.3 

52.8 

41.2 

68 

57.0 

37.0 

56.4 

38.0 

55.7 

39.0 

55.0 

40.0 

54.3 

40.9 

53.6 

41.9 

69 

57.9 

37.6 

57.2 

38.6 

56.5 

39.6 

55.8 

40.6 

55.1 

41.5 

54.4 

42.5 

70 

58.7 

38.1 

58.0 

39.1 

57.3 

40.2 

56.6 

41.1 

55.9 

42.1 

55.2 

43.1 

71 

59.5 

38.7 

58.9 

39.7 

58.2 

40.7 

57.4 

41.7 

56.7 

42.7 

55.9 

43.7 

72 

60.4 

39.2 

59.7 

40.3 

59.0 

41.3 

58.2 

42.3 

57.5 

43.3 

56.7 

44.3 

73 

61.2 

39.8 

60.5 

40.8 

59.8 

41.9 

59.1 

42.9 

58.3 

43.9 

57.5 

44.9 

74 

62.1 

40.3 

61.3 

41.4 

60.6 

42.4 

59.9 

43.5 

59.1 

44.5 

58.3 

45.6 

75 

62.9 

40.8 

62.2 

41.9 

61.4 

43.0 

60.7 

44.1 

59.9 

45.1 

59.1 

46.2 

76 

63.7 

41.4 

63.0 

42.5 

62.3 

43.6 

61.5 

44.7 

60.7 

45.7 

59.9 

46.8 

77 

64.6 

41.9 

63.8 

43.1 

63.1 

44.2 

62.3 

45.3 

61.5 

46.3 

60.7 

47.4 

78 

65.4 

42.5 

64.7 

43.6 

63.9 

44.7 

63.1 

45.8 

62.3 

46.9 

61.5 

48.0 

79 

66.3 

43.0 

65.5 

44.2 

64.7 

45.3 

63.9 

46.4 

63.1 

47.5 

62.3 

48.6 

80 

67.1 

43.6 

66.3 

44.7 

65.5 

45.9 

64.7 

47.0 

63.9 

48.1 

63.0 

49.3 

81 

67.9 

44.1 

67.2 

45.3 

66.4 

46.5 

65.5 

47.6 

64.7 

48.7 

63.8 

49.9 

82 

68.8 

44.7 

68.0 

45.9 

67.2 

47.0 

66.3 

48.2 

65.5 

49.3 

64.6 

50.5 

83 

69.6 

45.2 

68.8 

46.4 

68.0 

47.6 

67.1 

48.8 

66.3 

50.0 

65.4 

51.1 

84 

70.4 

45.7 

69.6 

47.0 

68.8 

48.2 

68.0 

49.4 

67.1 

50.6 

66.2 

51.7 

85 

71.3 

46.3 

70.5 

47.5 

69.6 

48.8 

68.8 

50.0 

67.9 

51.2 

67.0 

52.3 

86 

72.1 

46.8 

71.3 

48.1 

70.4 

49.3 

69.6 

50.5 

68.7 

51.8 

67.8 

52.9 

87 

73.0 

47.4 

72.1 

48.6 

71.3 

49.9 

70.4 

51.1 

69.5 

52.4 

68.6 

53.6 

88 

73.8 

47.9 

73.0 

49.2 

72.1 

50.5 

71.2 

51.7 

70.3 

53.0 

69.3 

54.2 

89 

74.6 

48.5 

73.8 

49.8 

72.9 

51.0 

72.0 

52.3 

71.1 

53.6 

70.1 

54.8 

90 

75.5 

49.0 

74.6 

50.3 

73.7 

51.6 

72.8 

52.9 

71.9 

54.2 

70.9 

55.4 

91 

76.3 

49.6 

75.4 

50.9 

74.5 

52.2 

73.6 

53.5 

72.7 

54.8 

71.7 

56.0 

92 

77.2 

50.1 

76.3 

51.4 

75.4 

52.8 

74.4 

54.1 

73.5 

55.4 

72.5 

56.6 

93 

78.0 

50.7 

77.1 

52.0 

76.2 

53.3 

75.2 

54.7 

74.3 

56.0 

73.3 

57.3 

94 

78.8 

51.2 

77.9 

52.6 

77.0 

53.9 

76.0 

55.3 

75.1 

56.6 

74.1 

57.9 

95 

79.7 

51.7 

78.8 

53.1 

77.8 

54.5 

76.9 

55.8 

75.9 

57.2 

74.9 

58.5 

96 

80.5 

52.3 

79.6 

53.7 

78.6 

55.1 

77.7 

56.4 

76.7 

57.8 

75.6 

59.1 

97 

81.4 

52.8 

80.4 

54.2 

79.5 

55.6 

78.5 

57.0 

77.5 

58.4 

76.4 

59.7 

98 

82.2 

53.4 

81.2 

54.8 

80.3 

56.2 

79.3 

57.6 

78.3 

59.0 

77.2 

60.3 

99 

83.0 

53.9 

82.1 

55.4 

81.1 

56.8 

80.1 

58.2 

79.1 

59.6 

78.0 

61.0 

100 

83.9 

54.5 

82.9 

55.9 

81.9 

57.4 

80.9 

58.8 

79.9 

60.2 

78.8 

61.6 

600 

503.2 

326.8 

497.4 

335.5 

491.5 

344.1 

485.4 

352.7 

479.2 

361.1 

472.8 

369.4 

700 

587.0 

381.3 

580.3 

391.4 

573.5 

401.5 

566.2 

411.4 

559.0 

421.3 

551.6 

430.8 

800 

671.0 

435.7 

663.3 

447.4 

655.4 

458.8 

647.3 

470.2 

638.9 

481.5 

630.4 

492.5 

900 

754.8 

490.1 

746.1 

503.2 

737.2 

516.2 

728.1 

528.9 

718.6 

541.7 

709.1 

554.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(123°,  237°, 

(124°,  236°, 

(125°,  235°, 

(126°,  234°, 

(127°,  233°, 

(128°,  232°, 

303°) 

304°) 

305°) 

306°) 

307°) 

308°) 

57° 

5  Pt.  56° 

55° 

54° 

4f  Pt,  53° 

52° 

The  5-Pt.  or  56°  Courses  are :  N.E.  by  E.,  S.E.  by  E.,  N.W.  by  W.,  S.W.  by  W. 


166 


Table  1.    Traverse  Table 


3J  Pt.  39° 

40° 

41° 

3f  Pt.  42° 

43° 

44° 

4  Pt.  45° 

DlST 

(141°,  219°, 
321°) 

(140°,  220°, 
320°) 

(139°,  221° 
319°) 

(138°,  222°, 
318°) 

(137°,  223°, 
317°) 

(136°,  224°, 
316°) 

(135°,  225°, 
315°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.8 

0.6 

0.8 

0.6 

0.8 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

2 

1.6 

1.3 

1.5 

1.3 

1.5 

1.3 

1.5 

1.3 

1.5 

1.4 

1.4 

1.4 

1.4 

1.4 

3 

2  'i 

1.9 

2.3 

1.9 

2.3 

2.0 

2.2 

2.0 

2.2 

2.0 

2.2 

2.1 

2.1 

2.1 

4 

3.1 

2.5 

3.1 

2.6 

3.0 

2.6 

3.0 

2.7 

2.9 

2.7 

2.9 

2.8 

2.8 

2.8 

5 

3.9 

3.1 

3.8 

3.2 

3.8 

3.3 

3.7 

3.3 

3.7 

3.4 

3.6 

3.5 

3.5 

3.5 

6 

4.7 

3.8 

4.6 

3.9 

4.5 

3.9 

4.5 

4.0 

4.4 

4.1 

4.3 

4.2 

4.2 

4.2 

7 

5.4 

4.4 

5.4 

4.5 

5.3 

4.6 

5.2 

4.7 

5.1 

4.8 

5.0 

4.9 

4.9 

4.9 

8 

6.2 

5.0 

6.1 

5.1 

6.0 

5.2 

5.9 

5.4 

5.9 

5.5 

5.8 

5.6 

5.7 

5.7 

9 

7.0 

5.7 

6.9 

5.8 

6.8 

5.9 

6.7 

6.0 

6.6 

6.1 

6.5 

6.3 

6.4 

6.4 

10 

7.8 

6.3 

7.7 

6.4 

7.5 

6.6 

7.4 

6.7 

7  3 

6.8 

7.2 

6.9 

7.1 

7.1 

11 

8.5 

6.9 

8.4 

7.1 

8.3 

7.2 

8.2 

7.4 

8.0 

7.5 

7.9 

7.6 

7.8 

7.8 

12 

9.3 

7.6 

9.2 

7.7 

9.1 

7.9 

8.9 

8.0 

8.8 

8.2 

8.6 

8.3 

8.5 

8.5 

13 

10.1 

8.2 

10.0 

8.4 

9.8 

8.5 

9.7 

8.7 

9.5 

8.9 

9.4 

9.0 

9.2 

9.2 

14 

10.9 

8.8 

10.7 

9.0 

10.6 

9.2 

10.4 

9.4 

10.2 

9.5 

10.1 

9.7 

9.9 

9.9 

15 

11.7 

9.4 

11.5 

9.6 

11.3 

9.8 

11.1 

10.0 

11.0 

10.2 

10.8 

10.4 

10.6 

10.6 

16 

12.4 

10.1 

12.3 

10.3 

12.1 

10.5 

11.9 

10.7 

11.7 

10.9 

11.5 

11.1 

11.3 

11.3 

17 

13.2 

10.7 

13.0 

10.9 

12.8 

11.2 

12.6 

11.4 

12.4 

11.6 

12  2 

11.8 

12.0 

12.0 

18 

14.0 

11.3 

13.8 

11.6 

13.6 

11.8 

13.4 

12.0 

13.2 

12.3 

12.9 

12.5 

12.7 

12.7 

19 

14.8 

12.0 

14.6 

12.2 

14.3 

12.5 

14.1 

12.7 

13.9 

13.0 

13.7 

13.2 

13.4 

13.4 

20 

15.5 

12.6 

15.3 

12.9 

15.1 

13.1 

14.9 

13.4 

14.6 

13.6 

14.4 

13.9 

14.1 

14.1 

21 

16.3 

13.2 

16.1 

13.5 

15.8 

13.8 

15.6 

14.1 

15.4 

14.3 

15.1 

14.6 

14.8 

14.8 

22 

17.1 

13.8 

16.9 

14.1 

16.6 

14.4 

16.3 

14.7 

16.1 

15.0 

15.8 

15.3 

15.6 

15.6 

23 

17.9 

14.5 

17.6 

14.8 

17.4 

15.1 

17.1 

15.4 

16.8 

15.7 

16.5 

16.0 

16.3 

16.3 

24 

18.7 

15.1 

18.4 

15.4 

18.1 

15.7 

17.8 

16.1 

17.6 

16.4 

17.3 

16.7 

17.0 

17.0 

25 

19.4 

15.7 

19.2 

16.1 

18.9 

16.4 

18.6 

16.7 

18.3 

17.0 

18.0 

17.4 

17.7 

17.7 

26 

20.2 

16.4 

19.9 

16.7 

19.6 

17.1 

19.3 

17.4 

19.0 

17.7 

18.7 

18.1 

18.4 

18.4 

27 

21.0 

17.0 

20.7 

17.4 

20.4 

17.7 

20.1 

18.1 

19.7 

18.4 

19.4 

18.8 

19.1 

19.1 

28 

21.8 

17.6 

21.4 

18.0 

21.1 

18.4 

20.8 

18.7 

20.5 

19.1 

20.1 

19.5 

19.8 

19.8 

29 

22.5 

18.3 

22.2 

18.6 

21.9 

19.0 

21.6 

19.4 

21.2 

19.8 

20.9 

20.1 

20.5 

20.5 

30 

23.3 

18.9 

23.0 

19.3 

22.6 

19.7 

22.3 

20.1 

21.9 

20.5 

21.6 

20.8 

21.2 

21.2 

31 

24.1 

19.5 

23.7 

19.9 

23.4 

20.3 

23.0 

20.7 

22.7 

21.1 

22.3 

21.5 

21.9 

21.9 

32 

24.9 

20.1 

24.5 

20.6 

24.2 

21.0 

23.8 

21.4 

23.4 

21.8 

23.0 

22.2 

22.6 

22.6 

33 

25.6 

20.8 

25.3 

21.2 

24.9 

21.6 

24.5 

22.1 

24.1 

22.5 

23.7 

22.9 

23.3 

23.3 

34 

26.4 

21.4 

26.0 

21.9 

25.7 

22.3 

25.3 

22.8 

24.9 

23.2 

24.5 

23.6 

24.0 

24.0 

35 

27.2 

22.0 

26.8 

22.5 

26.4 

23.0 

26.0 

23.4 

25.6 

23.9 

25.2 

24.3 

24.7 

24.7 

36 

28.0 

22.7 

27.6 

23.1 

27.2 

23.6 

26.8 

24.1 

26.3 

24.6 

25.9 

25.0 

25.5 

25.5 

37 

28.8 

23.3 

28.3 

23.8 

27.9 

24.3 

27.5 

24.8 

27.1 

25.2 

26.6 

25.7 

26.2 

26.2 

38 

29.5 

23.9 

29.1 

24.4 

28.7 

24.9 

28.2 

25.4 

27.8 

25.9 

27.3 

26.4 

26.9 

26.9 

39 

30.3 

24.5 

29.9 

25.1 

29.4 

25.6 

29.0 

26.1 

28.5 

26.6 

28.1 

27.1 

27.6 

27.6 

40 

31.1 

25.2 

30.6 

25.7 

30.2 

26.2 

29.7 

26.8 

29.3 

27.3 

28.8 

27.8 

28.3 

28.3 

41 

31.9 

25.8 

31.4 

26.4 

30.9 

26.9 

30.5 

27.4 

30.0 

28.0 

29.5 

28.5 

29.0 

29.0 

42 

32.6 

26.4 

32.2 

27.0 

31.7 

27.6 

31.2 

28.1 

30.7 

28.6 

30.2 

29.2 

29.7 

29.7 

43 

33.4 

27.1 

32.9 

27.6 

32.5 

28.2 

32.0 

28.8 

31.4 

29.3 

30.9 

29.9 

30.4 

30.4 

44 

34.2 

27.7 

33.7 

28.3 

33.2 

28.9 

32.7 

29.4 

32.2 

30.0 

31.7 

30.6 

31.1 

31.1 

45 

35.0 

28.3 

34.5 

28.9 

34.0 

29.5 

33.4 

30.1 

32.9 

30.7 

32.4 

31.3 

31.8 

31.8 

46 

35.7 

28.9 

35.2 

29.6 

34.7 

30.2 

34.2 

30.8 

33.6 

31.4 

33.1 

32.0 

32.5 

32.5 

47 

36.5 

29.6 

36.0 

30.2 

35.5 

30.8 

34.9 

31.4 

34.4 

32.1 

33.8 

32.6 

33.2 

33.2 

48 

37.3 

30.2 

36.8 

30.9 

36.2 

31.5 

35.7 

32.1 

35.1 

32.7 

34.5 

33.3 

33.9 

33.9 

49 

38.1 

30.8 

37.5 

31.5 

37.0 

32.1 

36.4 

32.8 

35.8 

33.4 

35.2 

34.0 

34.6 

34.6 

50 

38.9 

31.5 

38.3 

32.1 

37.7 

32.8 

37.2 

33.5 

36.6 

34.1 

36.0 

34.7 

35.4 

35.4 

100 

77.7 

62.9 

76.6 

64.3 

75.5 

65.6 

74.3 

66.9 

73.1 

68.2 

71.9 

69.5 

70.7 

70.7 

200 

55.4 

125.9 

53.2 

28.6 

50.9 

131.2 

48.6 

133.8 

46.3 

136.4 

143.9 

138.9 

141.4 

141.4 

300 

233.1 

188.8 

29.8 

92.8 

226.4 

196.8 

22.9 

200.7 

19.4 

204.6 

215.8 

208.4 

212.1 

212.1 

400 

310.9 

251.7 

06.4 

257.1 

301.9 

262.4 

97.3 

267.7 

92.6 

272.8 

287.7 

277.9 

282.8 

282.8 

500 

388.6 

314.7 

83.0 

321.4 

377.3 

328.0 

71.6 

334.6 

365.7 

341.0 

359.7 

347.3 

353.5 

353.5 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(129°,  231°, 

(130°,  230°, 

(131°,  229°, 

(132°,  228°, 

(133°,  227°, 

(134°,  226°, 

(135°,  225°, 

309°) 

310°) 

311° 

312°) 

313°) 

314°) 

315°) 

\\  Pt.  51° 

50° 

49° 

4i  Pt.  48° 

47° 

46° 

4  Pt.  45° 

The  4-Pt.  or  45°  Courses  are  :  N.E.,  N.W.,  S.E.,  S.W. 


Table  1.    Traverse  Table 


3^  Pt.  39° 

40° 

41° 

3f  Pt.42° 

43° 

44° 

4  Pt.  45° 

(141°,  219°, 

(140°,  220°, 

(139°,  221°, 

(138°,  222°, 

(137°,  223°, 

(136°,  224°, 

(135°,  225°, 

DlST. 

321°) 

320°) 

319°) 

318°) 

317°) 

316°) 

315°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

39.6 

32.1 

39.1 

32.8 

38.5 

33.5 

37.9 

34.1 

37.3 

34.8 

36.7 

35.4 

36.1 

36.1 

52 

40.4 

32.7 

39.8 

33.4 

39.2 

34.1 

38.6 

34.8 

38.0 

35.5 

37.4 

36.1 

36.8 

36.8 

53 

41.2 

33.4 

40.6 

34.1 

40.0 

34.8 

39.4 

35.5 

38.8 

36.1 

38.1 

36.8 

37.5 

37.5 

54 

42.0 

34.0 

41.4 

34.7 

40.8 

35.4 

40.1 

36.1 

39.5 

36.8 

38.8 

37.5 

38.2 

38.2 

55 

42.7 

34.6 

42.1 

35.4 

41.5 

36.1 

40.9 

36.8 

40.2 

37.5 

39.6 

38.2 

38.9 

38.9 

56 

43.5 

35.2 

42.9 

36.0 

42.3 

36.7 

41.6 

37.5 

41.0 

38.2 

40.3 

38.9 

39.6 

39.6 

57 

44.3 

35.9 

43.7 

36.6 

43.0 

37.4 

42.4 

38.1 

41.7 

38.9 

41.0 

39.6 

40.3 

40.3 

58 

45.1 

36.5 

44.4 

37.3 

43.8 

38.1 

43.1 

38.8 

42.4 

39.6 

41.7 

40.3 

41.0 

41.0 

59 

45.9 

37.1 

45.2 

37.9 

44.5 

38.7 

43.8 

39.5 

43.1 

40.2 

42.4 

41.0 

41.7 

41.7 

60 

46.6 

37.8 

46.0 

38.6 

45.3 

39.4 

44.6 

40.1 

43.9 

40.9 

43.2 

41.7 

42.4 

42.4 

61 

47.4 

38.4 

46.7 

39.2 

46.0 

40.0 

45.3 

40.8 

44.6 

41.6 

43.9 

42.4 

43.1 

43.1 

62 

48.2 

39.0 

47.5 

39.9 

46.8 

40.7 

46.1 

41.5 

45.3 

42.3 

44.6 

43.1 

43.8 

43.8 

63 

49.0 

39.6 

48.3 

40.5 

47.5 

41.3 

46.8 

42.2 

46.1 

43.0 

45.3 

43.8 

44.5 

44.5 

64 

49.7 

40.3 

49.0 

41.1 

48.3 

42.0 

47.6 

42.8 

46.8 

43.6 

46.0 

44.5 

45.3 

45.3 

65 

50.5 

40.9 

49.8 

41.8 

49.1 

42.6 

48.3 

43.5 

47.5 

44.3 

46.8 

45.2 

46.0 

46.0 

66 

51.3 

41.5 

50.6 

42.4 

49.8 

43.3 

49.0 

44.2 

48.3 

45.0 

47.5 

45.8 

46.7 

46.7 

67 

52.1 

42.2 

51.3 

43.1 

50.6 

44.0 

49.8 

44.8 

49.0 

45.7 

48.2 

46.5 

47.4 

47.4 

68 

52.8 

42.8 

52.1 

43.7 

51.3 

44.6 

50.5 

45.5 

49.7 

46.4 

48.9 

47.2 

48.1 

48.1 

69 

53.6 

43.4 

52.9 

44.4 

52.1 

45.3 

51.3 

46.2 

50.5 

47.1 

49.6 

47.9 

48.8 

48.8 

70 

54.4 

44.1 

53.6 

45.0 

52.8 

45.9 

52.0 

46.8 

51.2 

47.7 

50.4 

48.6 

49.5 

49.5 

71 

55.2 

44.7 

54.4 

45.6 

53.6 

46.6 

52.8 

47.5 

51.9 

48.4 

51.1 

49.3 

50.2 

50.2 

72 

56.0 

45.3 

55.2 

46.3 

54.3 

47.2 

53.5 

48.2 

52.7 

49.1 

51.8 

50.0 

50.9 

50.9 

73 

56.7 

45.9 

55.9 

46.9 

55.1 

47.9 

54.2 

48.8 

53.4 

49.8 

52.5 

50.7 

51.6 

51.6 

74 

57.5 

46.6 

56.7 

47.6 

55.8 

48.5 

55.0 

49.5 

54.1 

50.5 

53.2 

51.4 

52.3 

52.3 

75 

58.3 

47.2 

57.5 

48.2 

56.6 

49.2 

55.7 

50.2 

54.9 

51.1 

54.0 

52.1 

53.0 

53.0 

76 

59.1 

47.8 

58.2 

48.9 

57.4 

49.9 

56.5 

50.9 

55.6 

51.8 

54.7 

52.8 

53.7 

53.7 

77 

59.8 

48.5 

59.0 

49.5 

58.1 

50.5 

57.2 

51.5 

56.3 

52.5 

55.4 

53.5 

54.4 

54.4 

78 

60.6 

49.1 

59.8 

50.1 

58.9 

51.2 

58.0 

52.2 

57.0 

53.2 

56.1 

54.2 

55.2 

55.2 

79 

61.4 

49.7 

60.5 

50.8 

59.6 

51.8 

58.7 

52.9 

57.8 

53.9 

56.8 

54.9 

55.9 

55.9 

80 

62.2 

50.3 

61.3 

51.4 

60.4 

52.5 

59.5 

53.5 

58.5 

54.6 

57.5 

55.6 

56.6 

56.6 

81 

62.9 

51.0 

62.0 

52.1 

61.1 

53.1 

60.2 

54.2 

59.2 

55.2 

58.3 

56.3 

57.3 

57.3 

82 

63.7 

51.6 

62.8 

52.7 

61.9 

53.8 

60.9 

54.9 

60.0 

55.9 

59.0 

57.0 

58.0 

58.0 

83 

64.5 

52.2 

63.6 

53.4 

62.6 

54.5 

61.7 

55.5 

60.7 

56.6 

59.7 

57.7 

58.7 

58.7 

84 

65.3 

52.9 

64.3 

54.0 

63.4 

55.1 

62.4 

56.2 

61.4 

57.3 

60.4 

58.4 

59.4 

59.4 

85 

66.1 

53.5 

65.1 

54.6 

64.2 

55.8 

63.2 

56.9 

62.2 

58.0 

61.1 

59.0 

60.1 

60.1 

86 

66.8 

54.1 

65.9 

55.3 

64.9 

56.4 

63.9 

57.5 

62.9 

58.7 

61.9 

59.7 

60.8 

60.8 

87 

67.6 

54.8 

66.6 

55.9 

65.7 

57.1 

64.7 

58.2 

63.6 

59.3 

62.6 

60.4 

61.5 

61.5 

88 

68.4 

55.4 

67.4 

56.6 

66.4 

57.7 

65.4 

58.9 

64.4 

60.0 

63.3 

61.1 

62.2 

62.2 

89 

69.2 

56.0 

68.2 

57.2 

67.2 

58.4 

66.1 

59.6 

65.1 

60.7 

64.0 

61.8 

62.9 

62.9 

90 

69.9 

56.6 

68.9 

57.9 

67.9 

59.0 

66.9 

60.2 

65.8 

61.4 

64.7 

62.5 

63.6 

63.6 

91 

70.7 

57.3 

69.7 

58.5 

68.7 

59.7 

67.6 

60.9 

66.6 

62.1 

65.5 

63.2 

64.3 

64.3 

92 

71.5 

57.9 

70.5 

59.1 

69.4 

60.4 

68.4 

61.6 

67.3 

62.7 

66.2 

63.9 

65.1 

65.1 

93 

72.3 

58.5 

71.2 

59.8 

70.2 

61.0 

69.1 

62.2 

68.0 

63.4 

66.9 

64.6 

65.8 

65.8 

94 

73.1 

59.2 

72.0 

60.4 

70.9 

61.7 

69.9 

62.9 

68.7 

64.1 

67.6 

65.3 

66.5 

66.5 

95 

73.8 

59.8 

72.8 

61.1 

71.7 

62.3 

70.6 

63.6 

69.5 

64.8 

68.3 

66.0 

67.2 

67.2 

96 

74.6 

60.4 

73.5 

61.7 

72.5 

63.0 

71.3 

64.2 

70.2 

65.5 

69.1 

66.7 

67.9 

67.9 

97 

75.4 

61.0 

74.3 

62.4 

73.2 

63.6 

72.1 

64.9 

70.9 

66.2 

69.8 

67.4 

68.6 

68.6 

98 

76.2 

61.7 

75.1 

63.0 

74.0 

64.3 

72.8 

65.6 

71.7 

66.8 

70.5 

68.1 

69.3 

69.3 

99 

76.9 

62.3 

75.8 

63.6 

74.7 

64.9 

73.6 

66.2 

72.4 

67.5 

71.2 

68.8 

70.0 

70.0 

100 

77.7 

62.9 

76.6 

64.3 

75.5 

65.6 

74.3 

66.9 

73.1 

68.2 

71.9 

69.5 

70.7 

70.7 

600 

466.3 

377.6 

459.6 

385.7 

452.8 

393.6 

445.9 

401.5 

438.8 

409.2 

431.6 

416.8 

424.3 

424.3 

700 

543.9 

440.6 

536.3 

450.0 

528.3 

459.2 

520.2 

468.4 

511.9 

477.4 

503.5 

486.3 

495.0 

495.0 

800 

621.8 

503.5 

613.0 

514.2 

603.9 

524.8 

594.6 

535.3 

585.1 

545.6 

575.4 

555.8 

565.7 

565.7 

900 

699.3 

566.3 

689.5 

578.5 

679.2 

590.3 

668.8 

602.2 

658.2 

613.8 

647.3 

625.2 

636.3 

636.3 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(129°,  231°, 

(130°,  230°, 

(131°,  229°, 

(132°,  228°, 

(133°,  227°, 

(134°,  226°, 

(135°,  225°, 

309°) 

310°) 

311°) 

312°) 

313°) 

314°) 

315°) 

4|  Pt.  51° 

50° 

49° 

4J  Pt.  48° 

47° 

46° 

4  Pt.  45° 

The  4-Pt.  or  45°  Courses  are :  N.E.,  N.W.,  S.E.,  S.W. 


168 


Table  2 


To  CHANGE  LONG.  DIFF.  INTO  DEP.,  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIPF. 

OR 

MIDDLE  LATITUDE 

DEP. 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

1 

0.0 

0.0 

~bTo 

~o!b~ 

~ocT 

~ob~ 

~00 

~ob~ 

~QA 

"oo" 

~ob~ 

~ooT 

0.0 

0.0 

~ob~ 

2 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

3 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

o.i 

4 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

5 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

6 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

7 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0. 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

8 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

9 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0. 

0. 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

10 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0. 

0. 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

11 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0. 

0. 

0. 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

12 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

13 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0. 

0. 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

14 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

15 

0.0 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

16 

0.0 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

17 

0.0 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

0.6 

18 

0.0 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

19 

0.0 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.6 

20 

0.0 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

21 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

22 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

23 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

24 

0.0 

0.0 

0.0 

0. 

0.1 

0. 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

0.8 

25 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

0.9 

26 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

27 

0.0 

0.0 

0.0 

0. 

0. 

0. 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

28 

0.0 

0.0 

0.0 

0.1 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

.0 

29 

0.0 

0.0 

0.0 

0.1 

0. 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

0.9 

.0 

30 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

.0 

31 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

.1 

32 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

.1 

33 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

.1 

34 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

.2 

35 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.0 

.2 

36 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.7 

0.8 

0.9 

1.1 

.2 

37 

0.0 

0.0 

0. 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.1 

.3 

38 

0.0 

0.0 

0. 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

.0 

1.1 

1.3 

39 

0.0 

0.0 

0. 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

.0 

1.2 

1.3 

40 

0.0 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

.0 

1.2 

1.4 

41 

0.0 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

.1 

.2 

1.4 

42 

0.0 

0.0 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

.1 

.2 

1.4 

43 

0.0 

0.0 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

.1 

.3 

1.5 

44 

0.0 

0.0 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

.1 

.3 

1.5 

45 

0.0 

0.0 

0. 

0. 

0.2 

0.2 

0.3 

0.4 

0.6 

0.7 

0.8 

1.0 

.2 

.3 

.5 

46 

0.0 

0.0 

0.1 

0. 

0.2 

0.3 

0.3 

0.4 

0.6 

0.7 

0.8 

1.0 

.2 

.4 

.6 

47 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

.2 

1.4 

.6 

48 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

.2 

1.4 

.6 

49 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.1 

.3 

1.5 

.7 

50 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

.3 

1.5 

.7 

100 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.0 

3.4 

200 

0.0 

0.1 

0.3 

0.5 

0.8 

1.1 

1.5 

1.9 

2.5 

3.0 

3.7 

4.4 

5.1 

5.9 

6.8 

300 

0.0 

0.2 

0.4 

0.7 

1.1 

1.6 

2.2 

2.9 

3.7 

4.6 

5.5 

6.6 

7.7 

8.9 

10.2 

400 

0.1 

0.2 

0.6 

1.0 

1.5 

2.2 

3.0 

3.9 

4.9 

6.1 

7.4 

8.7 

10.2 

11.9 

13.7 

500 

0.1 

0.3 

0.7 

1.2 

1.9 

2.7 

3.7 

4.9 

6.2 

7.6 

9.2 

10.9 

12.8 

14.9 

17.0 

1.00 

1.00 

1.00 

1.00 

1.00 

1.01 

1.01 

1.01 

1.01 

1.02 

1.02 

1.02 

1.03 

1.03 

1.04 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.,  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN,  AND  ADD  PRODUCT  TO  DEP. 


Table  2 


To  CHANGE  LONG.  DIFF.  INTO  DEP.  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 

OR 

DEP. 

MIDDLE  LATITXTDE 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

51 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

1.3 

1.5 

1.7 

52 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1.0 

1.1 

1.3 

1.5 

1.8 

53 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

54 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

55 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

56 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0:7 

0.9 

1.0 

1.2 

1.4 

1.7 

1.9 

57 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.0 

1.2 

1.5 

1.7 

1.9 

58 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.7 

2.0 

59 

0.0 

0.0 

0.1 

0.1 

).2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.8 

2.0 

60 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.8 

2.0 

61 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

0.9 

1.1 

1.3 

1.6 

1.8 

2.1 

62 

0.0 

0.0 

0.1 

0.2 

0.2 

0.3 

0.5 

0.6 

0.8 

0.9 

1.1 

1.4 

1.6 

1.8 

2.1 

63 

0.0 

0.0 

0.1 

0.2 

0.2 

0.3 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

2.1 

64 

0.0 

0.0 

0.1 

0.2 

0.2 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

2.2 

65 

0.0 

0.0 

0.1 

0.2 

0.2 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.7 

1.9 

2.2 

66 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.7 

2.0 

2.2 

67 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.5 

1.7 

2.0 

2.3 

68 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.5 

1.7 

2.0 

2.3 

69 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

.0 

1.3 

1.5 

1.8 

2.0 

2.4 

70 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

.1 

1.3 

1.5 

1.8 

2.1 

2.4 

71 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

.1 

1.3 

1.6 

1.8 

2.1 

2.4 

72 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

.1 

1.3 

1.6 

1.8 

2.1 

2.5 

73 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

.1 

1.3 

1.6 

1.9 

2.2 

2.5 

74 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

.1 

1.4 

1.6 

1.9 

2.2 

2.5 

75 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

.1 

1.4 

1.6 

1.9 

2.2 

2.6 

76 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

.2 

1.4 

1.7 

1.9 

2.3 

2.6 

77 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

.2 

1.4 

1.7 

2.0 

2.3 

2.6 

78 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

.0 

.2 

1.4 

1.7 

2.0 

2.3 

2.7 

79 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

.0 

.2 

1.5 

1.7 

2.0 

2.3 

2.7 

80 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

.0 

.2 

1.5 

1.7 

2.1 

2.4 

2.7 

81 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

.0 

.2 

1.5 

1.8 

2.1 

2.4 

2.8 

82 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

.0 

.2 

1.5 

1.8 

2.1 

2.4 

2.8 

83 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

.0 

.3 

1.5 

1.8 

2.1 

2.5 

2.8 

84 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

.0 

.3 

1.5 

1.8 

2.2 

2.5 

2.9 

85 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

.0 

1.3 

1.6 

1.9 

2.2 

2.5 

2.9 

86 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

.1 

1.3 

1.6 

1.9 

2.2 

2.6 

2.9 

87 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.3 

1.6 

1.9 

2.2 

2.6 

3.0 

88 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.3 

1.6 

1.9 

2.3 

2.6 

3.0 

89 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.4 

1.6 

1.9 

2.3 

2.6 

3.0 

90 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

| 

1.4 

1.7 

2.0 

2.3 

2.7 

3.1 

91 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

.4 

"l.7 

2.0 

2.3 

2.7 

3.1 

92 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

< 

.4 

1.7 

2.0 

2.4 

2.7 

3.1 

93 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

.4 

1.7 

2.0 

2.4 

2.8 

3.2 

94 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

'.2 

.4 

1.7 

2.1 

2.4 

2.8 

3.2 

95 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

.2 

.4 

1.7 

2.1 

2.4 

2.8 

3.2 

96 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

.2 

.5 

1.8 

2.1 

2.5 

2.9 

3.3 

97 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

.2 

.5 

1.8 

2.1 

2.5 

2.9 

3.3 

98 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

.2 

.5 

1.8 

2.1 

2.5 

2.9 

3.3 

99 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

.2 

.5 

1.8 

2.2 

2.5 

2.9 

3.4 

100 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

.2 

.5 

1.8 

2.2 

2.6 

3.0 

3.4 

600 

0.1 

0.4 

0.8 

1.4 

2.3 

3.3 

4.5 

5.8 

7.4 

9.1 

10.0 

13.1 

15.4 

17.8 

20.5 

700 

0.2 

0.5 

1.0 

1.8 

2.8 

3.9 

5.1 

6.7 

8.7 

10.5 

12.9 

15.3 

17.9 

20.8 

23.9 

800 

0.2 

0.5 

1.1 

2.0 

3.1 

4.4 

5.9 

7.7 

9.8 

12.1 

14.8 

17.5 

20.6 

23.8 

27.3 

900 

0.3 

0.7 

1.4 

2.4 

3.6 

5.0 

6.7 

8.7 

11.2 

13.7 

16.7 

19.8 

23.2 

26.8 

30.8 

LOO 

LOO 

LOO 

LOO 

LOO 

L01 

L01 

1.01 

1.01 

1.02 

1.02 

1.02 

1.03 

1.03 

1.04 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


170 


Table 


To  CHANGE  LONG.  DIFF.  INTO  DEP.,  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 

OR 

MIDDLE  LATITUDE 

DEP. 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

1 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

2 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

3 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.4 

4 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

5 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

6 

0.2 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

7 

0.3 

0.3 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

8 

0.3 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

9 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

10 

0.4 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

11 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

12 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.4 

13 

0.5 

0.6 

0.6 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

14 

05 

0.6 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1  5 

1  6 

15 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.8 

16 

06 

0.7 

0.8 

0.9 

1.0 

1  i 

1  2 

1.3 

1.4 

1.5 

1  6 

1  7 

1  9 

17 

0.7 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

18 

0.7 

0.8 

0.9 

.0 

1.1 

1.2 

1.3 

1.4 

1:6 

1.7 

1.8 

2.0 

2.1 

19 

0.7 

0.8 

0.9 

.0 

1.1 

.3 

1.4 

1.5 

1.6 

1.8 

1.9 

2.1 

2.2 

20 

0.8 

0.9 

.0 

.1 

.2 

.3 

1.5 

1.6 

1.7 

1.9 

2.0 

2.2 

2.3 

21 

0.8 

0.9 

.0 

.1 

.3 

.4 

1.5 

1.7 

1.8 

2.0 

2.1 

2.3 

2.5 

22 

09 

.0 

.1 

.2 

.3 

.5 

1.6 

1.7 

1.9 

2.1 

2.2 

24 

9  6 

23 

0.9 

.0 

.1 

.3 

.4 

.5 

.  1.7 

1.8 

2.0 

2.2 

2.3 

2.5 

2.7 

24 

0.9 

.0 

.2 

.3 

.4 

.6 

1.7 

1.9 

2.1 

2.2 

2.4 

2.6 

2.8 

25 

1.0 

.1 

.2 

.4 

.5 

.7 

1.8 

2.0 

2.2 

2.3 

2.5 

2.7 

2.9 

26 

1.0 

.1 

.3 

.4 

.6 

.7 

1.9 

2.1 

2.2 

2.4 

2.6 

2.8 

3.0 

27 

.0 

.2 

.3 

.5 

.6 

1.8 

2.0 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

28 

.1 

.2 

.4 

.5 

.7 

1.9 

2.0 

2.2 

2.4 

2.6 

2.8 

3.1 

3.3 

29 

.1 

.3 

.4 

.6 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

3.4 

30 

.75 

.3 

.5 

.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.8 

3.0 

3.3 

3  5 

31 

.2 

.4 

.5 

.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.1 

3.4 

3.6 

32 

.2 

.4 

.6 

1.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.2 

3.5 

3.7 

33 

.3 

.4 

.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.3 

3.6 

3.9 

34 

.3 

.5 

.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

3.4 

3.7 

4.0 

35 

.4 

.5 

.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.3 

3.5 

3.8 

4.1 

36 

.4 

.6 

.8 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.4 

3.6 

3.9 

4.2 

37 

4 

.6 

1.8 

2.0 

2.2 

2.5 

2.7 

2.9 

3.2 

3.5 

3.7 

4.0 

43 

38 

5 

.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.3 

3.6 

3.8 

4.1 

44 

39 

5 

1.7 

1.9 

2.1 

2.4 

2.6 

2.8 

3.1 

3.4 

3.7 

3.9 

4.3 

4  6 

40 

5 

1.7 

2.0 

2.2 

2.4 

2.7 

2.9 

3.2 

3.5 

3.7 

4.0 

4.4 

47 

41 

.6 

1.8 

2.0 

2.2 

2.5 

2.7 

3.0 

3.3 

3.5 

3.8 

4.1 

4.5 

4.8 

42 

.6 

1.8 

2.1 

2.3 

2.5 

2.8 

3.1 

3.3 

3.6 

3.9 

4.3 

4.6 

4.9 

43 

.7 

1.9 

2.1 

2.3 

2.6 

2.9 

3.1 

3.4 

3.7 

4.0 

4.4 

4.7 

5.0 

44 

.7 

1.9 

2.2 

2.4 

2.7 

2.9 

3.2 

3.5 

3.8 

4.1 

4.5 

4.8 

5.2 

45 

.7 

2.0 

2.2 

2.5 

2.7 

3.0 

3.3 

3.6 

3.9 

4.2 

4.6 

4.9 

5.3 

46 

.8 

2.0 

2.3 

2.5 

2.8 

3.1 

3.3 

3.7 

4.0 

4.3 

4.7 

5.0 

5.4 

47 

.8 

2.1 

2.3 

2.6 

2.8 

3.1 

3.4 

3.7 

4.1 

4.4 

4.8 

5.1 

5.5 

48 

.9 

2.1 

2.3 

2.6 

2.9 

3.2 

3.5 

3.8 

4.1 

4.5 

4.9 

5.2 

5.6 

49 

,  .9 

2.1 

2.4 

2.7 

3.0 

3.3 

3.6 

3.9 

4.2 

4.6 

5.0 

5.3 

5.7 

50 

1.9 

2.2 

2.4 

2.7 

3.0 

3.3 

3.6 

4.0 

4.3 

4.7 

5.1 

5.4 

5.9 

100 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.7 

200 

7.7 

8.7 

9.8 

10.9 

12.1 

13.3 

14.6 

15.9 

17.3 

18.7 

20.2 

21.8 

23.4 

300 

11.6 

13.1 

14.7 

16.3 

18.1 

19.9 

21.8 

23.8 

25.9 

28.1 

30.4 

32.7 

35.1 

400 

15.5 

17.5 

19.6 

21.8 

24.1 

26.6 

29.1 

31.8 

34.6 

37.5 

40.5 

43.6 

46.9 

500 

19.4 

21.9 

24.5 

27.2 

30.1 

33.2 

36.4 

39.8 

43.2 

46.9 

50.6 

54.5 

58.5 

1.04 

1.05 

1.05 

1.06 

1.06 

1.07 

1.08 

1.09 

1.09 

1.10 

1.11 

1.12 

1.13 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.,  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


Table  2 


171 


To  CHANGE  LONG.  DIFF.  INTO  DEP.  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 

MIDDLE  LATITUDE 

OR 

DEP. 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

51 

2.0 

2.2 

2.5 

2.8 

3.1 

3.4 

3.7 

4.1 

4.4 

4.8 

5.2 

5.6 

6.0 

52 

2  0 

2  3 

2.5 

2.8 

3.1 

35 

3.8 

4.1 

4.5 

4.9 

5.3 

5.7 

6  1 

53 

2.1 

2.3 

2.6 

2.9 

3.2 

3.5 

3.9 

4.2 

4.6 

5.0 

5.4 

5.8 

6.2 

54 

2.1 

2.4 

2.6 

2.9 

3.3 

3.6 

3.9 

4.3 

4.7 

5.1 

5.5 

5.9 

6.3 

55 

2.1 

2.4 

2.7 

3.0 

3.3 

3.7 

4.0 

4.4 

4.8 

5.2 

5.6 

6.0 

6.4 

56 

2? 

2.4 

2.7 

3.1 

3.4 

3  7 

4.1 

4.5 

4.8 

5.2 

5.7 

6.1 

66 

57 

2  2 

2.5 

28 

3.1 

34 

3  8 

4.2 

4.5 

49 

5.3 

5.8 

6.2 

67 

58 

2.2 

2.5 

2.8 

3.2 

3.5 

3.9 

4.2 

4.6 

5.0 

5.4 

5.9 

6.3 

6.8 

59 

2.3 

2.6 

2.9 

3.2 

3.6 

3.9 

4.3 

4.7 

5.1 

5.5 

6.0 

6.4 

6.9 

60 

2.3 

2.6 

2.9 

3.3 

3.6 

4.0 

4.4 

4.8 

5.2 

5.6 

6.1 

6.5 

7.0 

61 

2.4 

2.7 

3.0 

3.3 

3.7 

4.1 

4.4 

4.8 

5.3 

5.7 

6.2 

6.6 

7.1 

62 

2.4 

2.7 

3.0 

3.4 

3.7 

4.1 

4.5 

4.9 

5.4 

5.8 

6.3 

6.8 

7.3 

63 

2.4 

2.8 

3.1 

3.4 

3.8 

4.2 

4.6 

5.0 

5.4 

5.9 

6.4 

6.9 

7.4 

64 

2.5 

2.8 

3.1 

3.5 

3.9 

4.3 

4.7 

5.1 

5.5 

6.0 

6.5 

7.0 

7.5 

65 

2.5 

2.8 

3.2 

3.5 

3.9 

4.3 

4.7 

5.2 

5.6 

6.1 

6.6 

7.1 

7.6 

66 

2.6 

2.9 

3.2 

3.6 

4.0 

4.4 

4.8 

5.2 

5.7 

6.2 

6.7 

7.2 

7.7 

67 

2.6 

2.9 

3.3 

3.7 

4.0 

4.5 

4.9 

5.3 

5.8 

6.3 

6.8 

7.3 

7.8 

68 

2.6 

3.0 

3.3 

3.7 

4.1 

4.5 

5.0 

5.4 

5.9 

6.4 

6.9 

7.4 

8.0 

69 

2.7 

3.0 

3.4 

3.8 

4.2 

4.6 

5.0 

5.5 

6.0 

6.5 

7.0 

7.5 

8.1 

70 

2.7 

3.1 

3.4 

3.8 

4.2 

4.6 

5.1 

5.6 

6.1 

6.6 

7.1 

7.6 

8.2 

71 

2.8 

3.1 

3.5 

3.9 

4.3 

4.7 

5.2 

5.6 

6.1 

6.7 

7.2 

7.7 

8.3 

72 

2.8 

3.1 

3.5 

3.9 

4.3 

4.8 

5.2 

5.7 

6.2 

6.7 

7.3 

7.8 

8.4 

73 

2.8 

3.2 

3.6 

4.0 

4.4 

4.8 

5.3 

5.8 

6.3 

6.8 

7.4 

8.0 

8.5 

74 

2.9 

3.2 

3.6 

4.0 

4.5 

4.9 

5.4 

5.9 

6.4 

6.9 

7.5 

8.1 

8.7 

75 

2.9 

3.3 

3.7 

4.1 

4.5 

5.0 

5.5 

6.0 

6.5 

7.0 

7.6 

8.2 

8.8 

76 

2.9 

3.3 

3.7 

4.1 

4.6 

5.0 

5.5 

6.0 

6.6 

7.1 

7.7 

8.3 

8.9 

77 

3.0 

3.4 

3.8 

4.2 

4.6 

5.1 

5.6 

6.1 

6.7 

7.2 

7.8 

8.4 

9.0 

78 

3.0 

3.4 

3.8 

4.2 

4.7 

5.2 

5.7 

6.2 

6.7 

7.3 

7.9 

8.5- 

9.1 

79 

3.1 

3.5 

3.9 

4.3 

4.8 

5.2 

5.8 

6.3 

6.8 

7.4 

8.0 

8.6 

9.2 

80 

3.1 

3.5 

3.9 

4.4 

4.8 

5.3 

5.8 

6.4 

6.9 

7.5 

8.1 

8.7 

9.4 

81 

3.1 

3.5 

4.0 

4.4 

4.9 

5.4 

5.9 

6.4 

7.0 

7.6 

8.2 

8.8 

9.5 

82 

3.2 

3.6 

4.0 

4.5 

4.9 

5.4 

6.0 

6.5 

7.1 

7.7 

8.3 

8.9 

9.6 

83 

3.2 

3.6 

4.1 

4.5 

5.0 

5.5 

6.0 

6.6 

7.2 

7.8 

8.4 

9.0 

9.7 

84 

3.3 

3.7 

4.1 

4.6 

5.1 

5.6 

6.1 

6.7 

7.3 

7.9 

8.5 

9.2 

9.8 

85 

3.3 

3.7 

4.2 

4.6 

5.1 

5.6 

6.2 

6.8 

7.3 

8.0 

8.6 

9.3 

9.9 

86 

3.3 

3.8 

4.2 

4.7 

5.2 

5.7 

6.3 

6.8 

7.4 

8.1 

8.7 

9.4 

10.1 

87 

3.4 

3.8 

4.3 

4.7 

5.2 

5.8 

6.3 

6.9 

7.5 

8.2 

8.8 

9.5 

10.2 

88 

3.4 

3.8 

4.3 

4.8 

5.3 

5.8 

6.4 

7.0 

7.6 

8.2 

8.9 

9.6 

10.3 

89 

3.4 

3.9 

4.4 

4.8 

5.4 

5.9 

6.5 

7.1 

7.7 

8.3 

9.0 

9.7 

10.4 

90 

3.5 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.2 

7.8 

8.4 

9.1 

9.8 

10.5 

91 

3.5 

4.0 

4.5 

5.0 

5.5 

6.0 

6.6 

7.2 

7.9 

8.5 

9.2 

9.9 

10.7 

92 

3.6 

4.0 

4.5 

5.0 

5.5 

6.1 

6.7 

7.3 

8.0 

8.6 

9.3 

10.0 

10.8 

93 

3.6 

4.1 

4.6 

5.1 

5.6 

6.2 

6.8 

7.4 

8.0 

8.7 

9.4 

10.1 

10.9 

94 

3.6 

4.1 

4.6 

5.1 

5.7 

6.2 

6.8 

7.5 

8.1 

8.8 

9.5 

10.2 

11.0 

95 

3.7 

4.2 

4.6 

5.2 

5.7 

6.3 

6.9 

7.6 

8.2 

8.9 

9.6 

10.4 

11.1 

96 

3.7 

4.2 

4.7 

5.2 

5.8 

6.4 

7.0 

7.6 

8.3 

9.0 

9.7 

10.5 

11.2 

97 

3.8 

4.2 

4.7 

5.3 

5.8 

6.4 

7.1. 

7.7 

8.4 

9.1 

9.8 

10.6 

11.4 

98 

3.8 

4.3 

4.8 

5.3 

5.9 

6.5 

7.1 

7.8 

8.5 

9.2 

9.9 

10.7 

11.5 

99 

3.8 

4.3 

4.8 

5.4 

6.0 

6.6 

7.2 

7.9 

8.6 

9.3 

10.0 

10.8 

11.6 

100 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.7 

600 

23.2 

26.2 

29.4 

32.7 

36.2 

39.9 

43.7 

47.7 

51.9 

56.2 

60.7 

65.4 

70.2 

700 

27.2 

30.6 

34.2 

38.1 

42.1 

46.4 

50.9 

55.7 

60.5 

65.5 

70.8 

76.3 

82.0 

800 

31.0 

35.0 

39.2 

43.5 

48.2 

53.1 

58.2 

63.6 

69.2 

74.9 

80.9 

87.1 

93.7 

900 

35.0 

39.4 

44.1 

49.1 

54.3 

59.7 

65.5 

71.7 

77.9 

84.4 

91.1 

98.1 

105.5 

1.04 

1.05 

1.05 

1.06 

1.06 

1.07 

1.08 

1.09 

1.10 

1.10 

1.11 

1.12 

1.13 

FACTOB 

To  CHANGE   DEP.  INTO   LONG.  DIFF.  MULTIPLY  T  ABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN,  AND  ADD  PRODUCT  TO  DEP. 


172 


Table  2 


To  CHANGE  LONG.  DIFF.  INTO  DEP.,  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 
OR 

MIDDLE  LATITUDE 

DEP. 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

4 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

5 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

1.2 

6 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

1.2 

1.3 

1.3 

1.4 

7 

0.9 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.3 

1.4 

1.5 

1.6 

1.6 

8 

.0 

.1 

1.1 

1.2 

1.3 

1.4 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

9 

.1 

.2 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

10 

.3 

.3 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

11 

.4 

.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.5 

2.6 

12 

.5 

.6 

1.7 

1.8 

1.9 

2.1 

2.2 

2.3 

2.4 

2.5 

2.7 

2.8 

13 

.6 

.7 

1.9 

2.0 

2.1 

2.2 

2.4 

2.5 

2.6 

2.8 

2.9 

3.0 

14 

.8 

.9 

2.0 

2.1 

2.3 

2.4 

2.5 

2.7 

2.8 

3.0 

3.1 

3.3 

15 

.9 

2.0 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.0 

3.2 

3.3 

3.5 

16 

2.0 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.7 

17 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4.0 

18 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.3 

3.4 

3.6 

3.8 

4.0 

4.2 

19 

2.4 

2.5 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.4 

20 

2.5 

2.7 

2.9 

3.0 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.5 

4.7 

21 

2.6 

2.8 

3.0 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.5 

4.7 

4.9 

22 

2.8 

2.9 

3.1 

3.3 

3.5 

3.8 

4.0 

4.2 

4.4 

4.7 

4.9 

5.1 

23 

2.9 

3.1 

3.3 

3.5 

3.7 

3,9 

4.2 

4.4 

4.6 

4.9 

5.1 

5.4 

24 

3.0 

3.2 

3.4 

3.6 

3.9 

4.1 

4.3 

4.6 

4.8 

5.1 

5.3 

5.6 

25 

3.1 

3.3 

3.6 

3.8 

4.0 

4.3 

4.5 

4.8 

5.0 

5.3 

5.6 

5.8 

26 

3.3 

3.5 

3.7 

4.0 

4.2 

4.4 

4.7 

5.0 

5.2 

5.5 

5.8 

6.1 

27 

3.4 

3.6 

3.9 

4.1 

4.4 

4.6 

4.9 

5.2 

5.4 

5.7 

6.0 

6.3 

28 

3.5 

3.8 

4.0 

4.3 

4.5 

4.8 

5.1 

5.3 

5.6 

5.9 

6.2 

6.6 

29 

3.6 

3.9 

4.1 

4.4 

4.7 

5.0 

5.2 

5.5 

5.8 

6.1 

6.5 

6.8 

30 

3.8 

4.0 

4.3 

4.6 

4.8 

5.1 

5.4 

5.7 

6.0 

6.4 

6.7 

7.0 

31 

3.9 

4.2 

4.4 

4.7 

5.0 

5.3 

5.6 

5.9 

6.2 

6.6 

6.9 

7.3 

32 

4.0 

4.3 

4.6 

4.9 

5.2 

5.5 

5.8 

6.1 

6.4 

6.8 

7.1 

7.5 

33 

4.1 

4.4 

4.7 

5.0 

5.3 

5.6 

6.0 

6.3 

6.6 

7.0 

7.4 

7.7 

34 

4.3 

4.6 

4.9 

5.2 

5.5 

5.8 

6.1 

6.5 

6.8 

7.2 

7.6 

8.0 

35 

4.4 

4.7 

5.0 

5.3 

5.6 

6.0 

6.3 

6.7 

7.0 

7.4 

7.8 

8.2 

36 

4.5 

4.8 

5.1 

5.5 

5.8 

6.2 

6.5 

6.9 

7.2 

7.6 

8.0 

8.4 

37 

4.6 

5.0 

5.3 

5.6 

6.0 

6.3 

6.7 

7.1 

7.5 

7.8 

8.2 

8.7 

38 

4.8 

5.1 

5.4 

5.8 

6.1 

6.5 

6.9 

7.3 

7.7 

8.1 

8.5 

8.9 

39 

4.9 

5.2 

5.6 

5.9 

6.3 

6.7 

7.1 

7.4 

7.9 

8.3 

8.7 

9.1 

40 

5.0 

5.4 

5.7 

6.1 

6.5 

6.8 

7.2 

7.6 

8.1 

8.5 

8.9 

9.4 

41 

5.1 

5.5 

5.9 

6.2 

6.6 

7.0 

7.4 

7.8 

8.3 

8.7 

9.1 

9.6 

42 

5.3 

5.6 

6.0 

6.4 

6.8 

7.2 

7.6 

8.0 

8.5 

8.9 

9.4 

9.8 

43 

5.4 

5.8 

6.1 

6.5 

6.9 

7.4 

7.8 

8.2 

8.7 

9.1 

9.6 

10.1 

44 

5.5 

5.9 

6.3 

6.7 

7.1 

7.5 

8.0 

8.4 

8.9 

9.3 

9.8 

10.3 

45 

5.6 

6.0 

6.4 

6.8 

7.3 

7.7 

8.1 

8.6 

9.1 

9.5 

10.0 

10.5 

46 

5.8 

6.2 

6.6 

7.0 

7.4 

7.9 

8.3 

8.8 

9.3 

9.8 

10.3 

10.8 

47 

5.9 

6.3 

6.7 

7.1 

7.6 

8.0 

8.5 

9.0 

9.5 

10.0 

10.5 

11.0 

48 

6.0 

6.4 

6.9 

7.3 

7.7 

8.2 

8.7 

9.2 

9.7 

10.2 

10.7 

11.2 

49 

6.1 

6.6 

7.0 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.4 

10.9 

11.5 

50 

6.3 

6.7 

7.1 

7.6 

8.1 

8.5 

9.0 

9.5 

10.1 

10.6 

11.1 

11.7 

100 

12.5 

13.4 

14.3 

15.2 

16.1 

17.1 

18.1 

19.1 

20.1 

21.2 

22.3 

23.4 

200 

25.1 

26.8 

28.6 

30.4 

32.3 

34.2 

36.2 

38.2 

40.3 

42.4 

44.6 

46.8 

300 

37.6 

40.2 

42.9 

45.6 

48.4 

51.3 

54.3 

57.3 

60.4 

63.6 

66.9 

70.2 

400 

50.2 

53.6 

57.1 

60.8 

64.5 

68.4 

72.3 

76.4 

80.6 

84.8 

89.1 

93.6 

500 

62.7 

67.0 

71.4 

76.0 

80.7 

85.5 

90.4 

95.5 

100.7 

106.0 

111.4 

117.0 

1.14 

1.15 

1.17 

1.18 

1.19 

1.21 

1.22 

1.24 

1.25 

1.27 

1.29 

1.31 

FACTOK 

To  CHANGE  DEP.  INTO  LONG.  DIFF.,  MULTIPLY  TABULAR  NUMBER   BY 
FACTOR  AT  FOOT  OF  COLUMN,  AND  ADD  PRODUCT  TO  DEP. 


Table  2 


To  CHANGE  LONG.  DIFF.  INTO  DEP.  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 

MIDDLE  LATITUDE 

OR 

DEP. 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

51 

6.4 

6.8 

7.3 

7.7 

8.2 

8.7 

9.2 

9.7 

10.3 

10.8 

11.4 

11.9 

52 

6.5 

7.0 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.5 

11.0 

11.6 

12.2 

53 

6.6 

7.1 

7.6 

8.1 

8.6 

9.1 

9.6 

10.1 

10.7 

11.2 

11.8 

12.4 

54 

6.8 

7.2 

7.7 

8.2 

8.7 

9.2 

9.8 

10.3 

10.9 

11.4 

12.0 

12.6 

55 

6.9 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.5 

11.1 

11.7 

12.3 

12.9 

56 

7.0 

7.5 

8.0 

8.5 

9.0 

9.6 

10.1 

10.7 

11.3 

11.9 

12.5 

13.1 

57 

7.1 

7.6 

8.1 

8.7 

9.2 

9.7 

10.3 

10.9 

11.5 

12.1 

12.7 

13.3 

58 

7.3 

7.8 

8.3 

8.8 

9.4 

9.9 

10.5 

11.1 

11.7 

12.3 

12.9 

13.6 

59 

7.4 

7.9 

8.4 

9.0 

9.5 

10.1 

10.7 

11.3 

11.9 

12.5 

13.1 

13.8 

60 

7.5 

8.0 

8.6 

9.1 

9.7 

10.3 

10.9 

11.5 

12.1 

12.7 

13.4 

14.0 

61 

7.6 

8.2 

8.7 

9.3 

9.8 

10.4 

11.0 

11.6 

12.3 

12.9 

13.6 

14.3 

62 

7.8 

8.3 

8.9 

9.4 

10.0 

10.6 

11.2 

11.8 

12.5 

13.1 

13.8 

14.5 

63 

7.9 

8.4 

9.0 

9.6 

10.2 

10.8 

11.4 

12.0 

12.7 

13.4 

14.0 

14.7 

64 

8.0 

8.6 

9.1 

9.7 

10.3 

10.9 

11.6 

12.2 

12.9 

13.6 

14.3 

15.0 

65 

8.1 

8.7 

9.3 

9.9 

10.5 

11.1 

11.8 

12.4 

13.1 

13.8 

14.5 

15.2 

66 

8.3 

8.8 

9.4 

10.0 

10.6 

11.3 

11.9 

12.6 

13.3 

14.0 

14.7 

15.4 

67 

8.4 

9.0 

9.6 

10.2 

10.8 

11.5 

12.1  ' 

12.8 

13.5 

14.2 

14.9 

15.7 

68 

8.5 

9.1 

9.7 

10.3 

11.0 

11.6 

12.3 

13.0 

13.7 

14.4 

15.2 

15.9 

69 

8.7 

9.2 

9.9 

10.5 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

15.4 

16.1 

70 

8.8 

9.4 

10.0 

10.6 

11.3 

12.0 

12.7 

13.4 

14.1 

14.8 

15.6 

16.4 

71 

8.9 

9.5 

10.1 

10.8 

11.5 

12.1 

12.8 

13.6 

14.3 

15.1 

15.8 

16.6 

72 

9.0 

9.6 

10.3 

10.9 

11.6 

12.3 

13.0 

13.8 

14.5 

15.3 

16.0 

16.8 

73 

9.2 

9.8 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.7 

15.5 

16.3 

17.1 

74 

9.3 

9.9 

10.6 

11.2 

11.9 

12.7 

13.4 

14.1 

14.9 

15.7 

16.5 

17.3 

75 

9.4 

10.0 

10.7 

11.4 

12.1 

12.8 

13.6 

14.3 

15.1 

15.9 

16.7 

17.5 

76 

9.5 

10.2 

10.9 

11.5 

12.3 

13.0 

13.7 

14.5 

15.3 

16.1 

16.9 

17.8 

77 

9.7 

10.3 

11.0 

11.7 

12.4 

13.2 

13.9 

14.7 

15.5 

16.3 

17.2 

18.0 

78 

98 

10.5 

11.1 

11.9 

12.6 

13.3 

14.1 

14.9 

15.7 

16.5 

174 

182 

79 

9.9 

10.6 

11.3 

12.0 

12.7 

13.5 

14.3 

15.1 

15.9 

16.7 

17.6 

18.5 

80 

10.0 

10.7 

11.4 

12.2 

12.9 

13.7 

14.5 

15.3 

16.1 

17.0 

17.8 

18.7 

81 

10.2 

10.9 

11.6 

12.3 

13.1 

13.8 

14.6 

15.5 

16.3 

17.2 

18.1 

19.0 

82 

10.3 

11.0 

11.7 

12.5 

13.2 

14.0 

14.8 

15.7 

16.5 

17.4 

18.3 

19.2 

83 

10.4 

11.1 

11.9 

12.6 

13.4 

14.2 

15.0 

15.9 

16.7 

17.6 

18.5 

19.4 

84 

10.5 

11.3 

12.0 

12.8 

13.6 

14.4 

15.2 

16.0 

16.9 

17.8 

18.7 

19.7 

85 

10.7 

11.4 

12.1 

12.9 

13.7 

14.5 

15.4 

16.2 

17.1 

18.0 

18.9 

19.9 

86 

10.8 

11.5 

12.3 

13.1 

13.9 

14.7 

15.6 

16.4 

17.3 

18.2 

19.2 

20.1 

87 

10.9 

11.7 

12.4 

13.2 

14.0 

14.9 

15.7 

16.6 

17.5 

18.4 

19.4 

20.4 

88 

11.0 

11.8 

12.6 

13.4 

14.2 

15.0 

15.9 

16.8 

17.7 

18.7 

19.6 

20.6 

89 

11.2 

11.9 

12.7 

13.5 

14.4 

15.2 

16.1 

17.0 

17.9 

18.9 

19.8 

20.8 

90 

11.3 

12.1 

12.9 

13.7 

14.5 

15.4 

16.3 

17.2 

18.1 

19.1 

20.1 

21.1 

91 

11.4 

12.2 

13.0 

13.8 

14.7 

15.6 

16.5 

17.4 

18.3 

19.3 

20.3 

21.3 

92 

11.5 

12.3 

13.1 

14.0 

14.8 

15.7 

16.6 

17.6 

18.5 

19.5 

20.5 

21.5 

93 

11.7 

12.5 

13.3 

14.1 

15.0 

15.9 

16.8 

17.8 

18.7 

19.7 

20.7 

21.8 

94 

11.8 

12.6 

13.4 

14.3 

15.2 

16.1 

17.0 

18.0 

18.9 

19.9 

20.9 

22.0 

95 

11.9 

12.7 

13.6 

14.4 

15.3 

16.2 

17.2 

18.1 

19.1 

20.1 

21.2 

22.2 

96 

12.0 

12.9 

13.7 

14.6 

15.5 

16.4 

17.4 

18.3 

19.3 

20.4 

21.4 

22.5 

97 

12.2 

13.0 

13.9 

14.7 

15.6 

16.6 

17.5 

18.5 

19.5 

20.6 

21.6 

22.7 

98 

12.3 

13.1 

14.0 

14.9 

15.8 

16.8 

17.7 

18.7 

19.7 

20.8 

21.8 

22.9 

99 

12.4 

13.3 

14.1 

15.0 

16.0 

16.9 

17.9 

18.9 

19.9 

21.0 

22.1 

23.2 

100 

12.5 

13.4 

14.3 

15.2 

16.1 

17.1 

18.1 

19.1 

20.1 

21.2 

22.3 

23.4 

600 

75.2 

80.4 

85.7 

91.2 

96.8 

102.6 

108.5 

114.6 

120.8 

127.2 

133.7 

140.4 

700 

87.8 

93.9 

99.9 

106.4 

113.0 

119.7 

126.5 

133.8 

141.0 

148.4 

156.1 

163.7 

800 

100.3 

107.2 

114.2 

121.6 

129.0 

136.7 

144.6 

152.7 

161.1 

169.6 

178.2 

187.0 

900 

113.0 

120.7 

128.6 

136.8 

145.2 

153.9 

162.8 

171.9 

181.4 

190.9 

200.7 

210.5 

1.14 

1.15 

1.17 

1.18 

1.19 

1.21 

1.22 

1.24 

1.25 

1.27 

1.29 

1.31 

!                              FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


174 


Table  2 


To   CHANGE    LONG.   DIFP.   INTO   DEP.,   SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIPF. 

OR 

MIDDLE  LATITUDE 

DEP. 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

1 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

2 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.7 

3 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.0 

1.1 

1.1 

4 

1.0 

1.0 

1.1 

1.1 

1.2 

1.2 

1.3 

1.3 

1.4 

1.4 

15 

5 

1.2 

1.3 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.7 

1.8 

1.9 

6 

1.5 

1.5 

1.6 

1.7 

1.8 

1.8 

1.9 

2.0 

2.1 

2.1 

2.2 

7 

1.7 

1.8 

1.9 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

8 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.8 

2.9 

3.0 

9 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

2.9 

3.0 

3.1 

3.2 

3.3 

10 

2.5 

2.6 

2.7 

2.8 

2.9 

3.1 

3.2 

3.3 

3.4 

3.6 

3.7 

11 

2.7 

2.8 

3.0 

3.1 

3.2 

3.4 

3.5 

3.6 

3.8 

3.9 

4.1 

12 

2.9 

3.1 

3.2 

3.4 

3.5 

3.7 

3.8 

4.0 

4.1 

4.3 

4.4 

13 

3.2 

3.3 

3.5 

3.6 

3.8 

4.0 

4.1 

4.3 

4.5 

4.6 

4.8 

14 

3.4 

3.6 

3.8 

3.9 

4.1 

4.3 

4.5 

4.6 

4.8 

5.0 

5.2 

15 

3.7 

3.9 

4.0 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

5.4 

5.6 

16 

3.9 

4.1 

4.3 

4.5 

4.7 

4.9 

5.1 

5.3 

5.5 

5.7 

5.9 

17 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

5.4 

5.6 

5.8 

6.1 

6.3 

18 

4.4 

4.6 

4.8 

5.1 

5.3 

5.5 

5.7 

6.0 

6.2 

6.4 

6.7 

19 

4.7 

4.9 

5.1 

5.3 

5.6 

5.8 

6.0 

6.3 

6.5 

6.8 

7.0 

20 

4.9 

5.1 

5.4 

5.6 

5.9 

6.1 

6.4 

6.6 

6.9 

7.1 

7.4 

21 

5.2 

5.4 

5.6 

5.9 

6.2 

6.4 

6.7 

6.9 

7.2 

7.5 

7.8 

22 

5.4 

5.7 

5.9 

6.2 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.2 

23 

5.6 

5.9 

6.2 

6.5 

6.7 

7.0 

7.3 

7.6 

7.9 

8.2 

8.5 

24 

5.9 

6.2 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

25 

6.1 

6.4 

6.7 

7.0 

7.3 

7.6 

8.0 

8.3 

8.6 

8.9 

9.3 

26 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

9.3 

9.6 

27 

6.6 

6.9 

7.3 

7.6 

7.9 

8.2 

8.6 

8.9 

9.3 

9.6 

10.0 

28 

6.9 

7.2 

7.5 

7.9 

8.2 

8.5 

8.9 

9.3 

9.6 

10.0 

10.4 

29 

7.1 

7.4 

7.8 

8.1 

8.5 

8.9 

9.2 

9.6 

10.0 

10.4 

10.7 

30 

7.4 

7.7 

8.1 

8.4 

8.8 

9.2 

9.5 

9.9 

10.3 

10.7 

11.1 

31 

.  7.6 

8.0 

8.3 

8.7 

9.1 

9.5 

9.9 

10.3 

10.7 

11.1 

11.5 

32 

7.8 

8.2 

8.6 

9.0 

9.4 

9.8 

10.2 

10.6 

11.0 

11.4 

11.9 

33 

8.1 

8.5 

8.9 

9.3 

9.7 

10.1 

10.5 

10.9 

11.4 

11.8 

12.2 

34 

8.3 

8.7 

9.1 

9.5 

10.0 

10.4 

10.8 

11.2 

11.7 

12.1 

12.6 

35 

8.6 

9.0 

9.4 

9.8 

10.3 

10.7 

11.1 

11.6 

12.0 

12.5 

13.0 

36 

8.8 

9.2 

9.7 

10.1 

10.5 

11.0 

11.4 

11.9 

12.4 

12.9 

13.3 

37 

9.1 

9.5 

9.9 

10.4 

10.8 

11.3 

11.8 

12.2 

12.7 

13.2 

13.7 

38 

9  3 

9.8 

10.2 

10.7 

11.1 

11.6 

12.1 

12.6 

13.1 

13.6 

14  1 

39 

9  6 

10  0 

10.5 

10.9 

11.4 

11.9 

12.4 

12.9 

13.4 

13.9 

145 

40 

98 

10.3 

107 

11.2 

11.7 

12.2 

12.7 

13.2 

13.8 

14.3 

148 

41 

10.1 

10.5 

11.0 

11.5 

12.0 

12.5 

13.0 

13.6 

14.1 

14.6 

15.2 

42 

10.3 

10.8 

11.3 

11.8 

12.3 

12.8 

13.4 

13.9 

14.4 

15.0 

15.6 

43 

10.5 

11.0 

11.6 

12.1 

12.6 

13.1 

13.7 

14.2 

14.8 

15.4 

15.9 

44 

10.8 

11.3 

11.8 

12.3 

12.9 

13.4 

14.0 

14.6 

15.1 

15.7 

16.3 

45 

11.0 

11.6 

12.1 

12.6 

13.2 

13.7 

14.3 

14.9 

15.5 

16.1 

16.7 

46 

11.3 

11.8 

12.4 

12.9 

13.5 

14.0 

14.6 

15.2 

15.8 

16.4 

17.1 

47 

11.5 

12.1 

12.6 

13.2 

13.8 

14.4 

14.9 

15.6 

16.2 

16.8 

17.4 

48 

11.8 

12.3 

12.9 

13.5 

14.1 

14.7 

15.3 

15.9 

16.5 

17.1 

17.8 

49 

12.0 

12.6 

13.2 

13.8 

14.4 

15.0 

15.6 

16.2 

16.9 

17.5 

18.2 

50 

12.3 

12.8 

13.4 

14.0 

14.6 

15.3 

15.9 

16.5 

17.2 

17.9 

18.5 

100 

24.5 

25.7 

26.9 

28.1 

29.3 

30.5 

31.8 

33.1 

34.4 

35.7 

37.1 

200 

49.1 

51.4 

53.7 

56.1 

58.6 

61.1 

63.6 

66.2 

68.8 

71.4 

74.1 

300 

73.6 

77.1 

80.6 

84.2 

87.9 

91.6 

95.4 

99.3 

103..2 

107.2 

111.2 

400 

98.1 

102.7 

107.4 

112.3 

117.2 

122.1 

127.2 

132.3 

137.6 

142.9 

148.3 

500 

122.7 

128.4 

134.3 

140.3 

146.5 

152.7 

159.0 

165.4 

172.0 

178.6 

185.3 

1.33 

1.35 

1.37 

1.39 

1.41 

1.44 

1.47 

1.50 

1.52 

1.56 

1.59 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.,  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN,  AND  ADD  PRODUCT  TO  DEP. 


Table  2 


175 


To  CHANGE  LONG.  DIFF.  INTO  DEP.  SUBTRACT  TABULAE  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF 

OR 

MIDDLE  LATITUDE 

DEP. 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

51 

12.5 

13.1 

13.7 

14.3 

14.9 

15.6 

16.2 

16.9 

17.5 

18.2 

18.9 

52 

12.8 

13.4 

14.0 

14.6 

15.2 

15.9 

16.5 

17.2 

17.9 

18.6 

19.3 

53 

13.0 

13.6 

14.2 

14.9 

15.5 

16.2 

16.9 

17.5 

18.2 

18.9 

19.6 

54 

13.2 

13.9 

14.5 

15.2 

15.8 

16.5 

17.2 

17.9 

18.6 

19.3 

20.0 

55 

13.5 

14.1 

14.8 

15.4 

16.1 

16.8 

17.5 

18.2 

18.9 

19.6 

20.4 

56 

13.7 

14.4 

15.0 

15.7 

16.4 

17.1 

17.8 

18.5 

19.3 

20.0 

20.8 

57 

14.0 

14.6 

15.3 

16.0 

16.7 

17.4 

18.1 

18.9 

19.6 

20.4 

21.1 

58 

14.2 

14.9 

15.6 

16.3 

17.0 

17.7 

18.4 

19.2 

19.9 

20.7 

21.5 

59 

14.5 

15.2 

15.9 

16.6 

17.3 

18.0 

18.8 

19.5 

20.3 

21.1 

21.9 

60 

14.7 

15.4 

16.1 

16.8 

17.6 

18.3 

19.1 

19.9 

20.6 

21.4 

22.2 

61 

15.0 

15.7 

16.4 

17.1 

17.9 

18.6 

19.4 

20.2 

21.0 

21.8 

22.6 

62 

15.2 

15.9 

16.7 

17.4 

18.2 

18.9 

19.7 

20.5 

21.3 

22.1 

23.0 

63 

15.5 

16.2 

16.9 

17.7 

18.5 

19.2 

20.0 

20.8 

21.7 

22.5 

23.4 

64 

15.7 

16.4 

17.2 

18.0 

18.7 

19.5 

20.4 

21.2 

22.0 

22.9 

23.7 

65 

15.9 

16.7 

17.5 

18.2 

19.0 

19.8 

20.7 

21.5 

22.4 

23.2 

24.1 

66 

16.2 

17.0 

17.7 

18.5 

19.3 

20.2 

21.0 

21.8 

22.7 

23.6 

24.5 

67 

16.4 

17.2 

18.0 

18.8 

19.6 

20.5 

21.3 

22.2 

23.0 

23.9 

24.8 

68 

16.7 

17.5 

18.3 

19.1 

19.9 

20.8 

21.6 

22.5 

23.4 

24.3 

25.2 

69 

16.9 

17.7 

18.5 

19.4 

20.2 

21.1 

21.9 

22.8 

23.7 

24.6 

25.6 

70 

17.2 

18.0 

18.8 

19.6 

20.5 

21.4 

22.3 

23.2 

24.1 

25.0 

25.9 

71 

17.4 

18.2 

19.1 

19.9 

20.8 

21.7 

22.6 

23.5 

24.4 

25.4 

26.3 

72 

17.7 

18.5 

19.3 

20.2 

21.1 

22.0 

22.9 

23.8 

24.8 

25.7 

26.7 

73 

17.9 

18.8 

19.6 

20.5 

21.4 

22.3 

23.2 

24.2 

25.1 

26.1 

27.1 

74 

18.2 

19.0 

19.9 

20.8 

21.7 

22.6 

23.5 

24.5 

25.5 

26.4 

27.4 

75 

18.4 

19.3 

20.1 

21.0 

22.0 

22.9 

23.9 

24.8 

25.8 

26.8 

27.8 

76 

18.6 

19.5 

20.4 

21.3 

22.3 

23.2 

24.2 

25.1 

26.1 

27.1 

28.2 

77 

18.9 

19.8 

20.7 

21.6 

22.6 

23.5 

24.5 

25.5 

26.5 

27.5 

28.5 

78 

19.1 

20.0 

21.0 

21.9 

22.8 

23.8 

24.8 

25.8 

26.8 

27.9 

28.9 

79 

19.4 

20.3 

21.2 

22.2 

23.1 

24.1 

25.1 

26.1 

27.2 

28.2 

29.3 

80 

19.6 

20.5 

21.5 

22.5 

23.4 

24.4 

25.4 

26.5 

27.5 

28.6 

29.7 

81 

19.9 

20.8 

21.8 

22.7 

23.7 

24.7 

25.8 

26.8 

27.9 

28.9 

30.0 

82 

20.1 

21.1 

22.0 

23.0 

24.0 

25.0 

26.1 

27.1 

28.2 

29.3 

30.4 

83 

20.4 

21.3 

22.3 

23.3 

24.3 

25.3 

26.4 

27.5 

28.5 

29.6 

30.8 

84 

20.6 

21.6 

22.6 

23.6 

24.6 

25.6 

26.7 

27.8 

28.9 

30.0 

31.1 

85 

20.8 

21.8 

22.8 

23.9 

24.9 

26.0 

27.0 

28.1 

29.2 

30.4 

31.5 

86 

21.1 

22.1 

23.1 

24.1 

25.2 

26.3 

27.3 

28.5 

29.6 

30.7 

31.9 

87 

21.3 

22.3 

23.4 

24  .4 

25.5 

26.6 

27.7 

28.8 

29.9 

31.1 

32.2 

88 

21.6 

22.6 

23.6 

24.7 

25.8 

26.9 

28.0 

29.1 

30.3 

31.4 

32.6 

89 

21.8 

22.9 

23.9 

25.0 

26.1 

27.2 

28.3 

29.4 

30.6 

31.8 

33.0 

90 

22.1 

23.1 

24.2 

25.3 

26.4 

27.5 

28.6 

29.8 

31.0 

32.1 

33.4 

91 

22.3 

23.4 

24.4 

25.5 

26.7 

27.8 

28.9 

30.1 

31.3 

32.5 

33.7 

92 

22.6 

23.6 

24.7 

25.8 

26.9 

28.1 

29.3 

30.4 

31.6 

32.9 

34.1 

93 

22.8 

23.9 

25.0 

26.1 

27.2 

28.4 

29.6 

30.8 

32.0 

33.2 

34.5 

94 

23.1 

24.1 

25.3 

26.4 

27.5 

28.7 

29.9 

31.1 

32.3 

33.6 

34.8 

95 

23.3 

24.4 

25.5 

26.7 

27.8 

29.0 

30.2 

31.4 

32.7 

33.9 

35.2 

96 

23.5 

24.7 

25.8 

26.9 

28.1 

29.3 

30.5 

31.8 

33.0 

34.3 

35.6 

97 

23.8 

24.9 

26.1 

27.2 

28.4 

29.6 

30.8 

32.1 

33.4 

34.6 

36.0 

98 

24.0 

25.2 

26.3 

27.5 

28.7 

29.9 

31.2 

32.4 

33.7 

35.0 

36.3 

99 

24.3 

25.4 

26.6 

27.8 

29.0 

30.2 

31.5 

32.8 

34.1 

35.4 

36.7 

100 

24.5 

25.7 

26.9 

28.1 

29.3 

30.5 

31.8 

33.1 

34.4 

35.7 

37.1 

600 

147.2 

154.1 

161.2 

168.4 

175.7 

183.2 

190.8 

198.5 

206.4 

214.3 

222.4 

700 

171.7 

179.8 

188.1 

196.5 

205.0 

213.7 

222.6 

231.6 

240.8 

250.0 

259.4 

800 

196.1 

205.4 

214.9 

224.6 

234.3 

244.2 

254.4 

264.7 

275.2 

285.8 

296.5 

900 

220.8 

231.2 

241.8 

252.7 

263.7 

274.8 

286.2 

297.8 

309.7 

321.5 

333.7 

1.33 

1.35 

1.37 

1.39 

1.41 

1.44 

1.47 

1.50 

1.52 

1.56 

1.59 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


176 


Table  2 


To  CHANGE  LONG.  DIFF.  INTO  DEP.,  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFP. 

OH 

MIDDLE  LATITUDE 

DEP. 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

1 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.5 

2 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.9 

1.0 

1.0 

3 

1.2 

1.2 

1.2 

1.3 

1.3 

1.4 

1.4 

1.5 

1.5 

4 

1.5 

1.6 

1.6 

1.7 

1.8 

1.8 

1.9 

1.9 

2.0 

5 

1.9 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.4 

2.5 

6 

2.3 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.9 

3.0 

7 

2.7 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

8 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.8 

3.9 

4.0 

9 

3.5 

3.6 

3.7 

3.8 

4.0 

4.1 

4.2 

4.4 

4.5 

10 

3.8 

4.0 

4.1 

4.3 

4.4 

4.6 

4.7 

4.8 

5.0 

11 

4.2 

4.4 

4.5 

4.7 

4.8 

5.0 

5.2 

5.3 

5.5 

12 

4.6 

4.8 

4.9 

5.1 

5.3 

5.5 

5.6 

5.8 

6.0 

13 

5.0 

5.2 

5.4 

5.5 

5.7 

5.9 

6.1 

6.3 

6.5 

14 

5.4 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.8 

7.0 

15 

5.8 

6.0 

6.2 

6.4 

6.6 

6.8 

7.1 

7.3 

7.5 

16 

6.1 

6.4 

6.6 

6.8 

7.1 

7.3 

7.5 

7.8 

8.0 

17 

6.5 

6.8 

7.0 

7.2 

7.5 

7.7 

8.0 

8.2 

8.5 

18 

6.9 

7.2 

7.4 

7.7 

7.9 

8.2 

8.5 

8.7 

9.0 

19 

7.3 

7.6 

7.8 

8.1 

8.4 

8.7 

8.9 

9.2 

9.5 

20 

7.7 

8.0 

8.2 

8.5 

8.8 

9.1 

9.4 

9.7 

10.0 

21 

8.1 

8.4 

8.7 

9.0 

9.3 

9.6 

9.9 

10.2 

10.5 

22 

8.5 

8.8 

9.1 

9.4 

9.7 

10.0 

10.3 

10.7 

11.0 

23 

8.8 

9.2 

9.5 

9.8 

10.1 

10.5 

10.8 

11.2 

11.5 

24 

9.2 

9.6 

9.9 

10.2 

10.6 

10.9 

11.3 

11.6 

12.0 

25 

9.6 

10.0 

10.3 

10.7 

11.0 

11.4 

11.8 

12.1 

12.5 

26 

10.0 

10.4 

10.7 

11.1 

11.5 

11.8 

12.2 

12.6 

13.0 

27 

10.4 

10.8 

11.1 

11.5 

11.9 

12.3 

12.7 

13.1 

13.5 

28 

10.8 

11.1 

11.5 

11.9 

12.3 

12.8 

13.2 

13.6 

14.0 

29 

11.1 

11.5 

12.0 

12.4 

12.8 

13.2 

13.6 

14.1 

14.5 

30 

11.5 

11.9 

12.4 

12.8 

13.2 

13.7 

14.1 

14.5 

15.0 

31 

11.9 

12.3 

12.8 

13.2 

13.7 

14.1 

14.6 

15.0 

15.5 

32 

12.3 

12.7 

13.2 

13.6 

14.1 

14.6 

15.0 

15.5 

16.0 

33 

12.7 

13.1 

13.6 

14.1 

14.5 

15.0 

15.5 

16.0 

16.5 

34 

13.1 

13.5 

14.0 

14.5 

15.0 

15.5 

16.0 

16.5 

17.0 

35 

13.5 

13.9 

14.4 

14.9 

15.4 

15.9 

16.5 

17.0 

17.5 

36 

13.8 

14.3 

14.8 

15.4 

15.9 

16.4 

16.9 

17.5 

18.0 

37 

14.2 

14.7 

15.3 

15.8 

16.3 

16.8 

17.4 

17.9 

18.5 

38 

14.6 

15.1 

15.7 

16.2 

16.8 

17.3 

17.9 

18.4 

19.0 

39 

15.0 

15.5 

16.1 

16.6 

17.2 

17.8 

18.3 

18.9 

19.5 

40 

15.4 

15.9 

16.5 

17.1 

17.6 

18.2 

18.8 

19.4 

20.0 

41 

15.8 

16.3 

16.9 

17.5 

18.1 

18.7 

19.3 

19.9 

20.5 

42 

16.1 

16.7 

17.3 

17.9 

18.5 

19.1 

19.7 

20.4 

21.0 

43 

16.5 

17.1 

17.7 

18.3 

19.0 

19.6 

20.2 

20.9 

21.5 

44 

16.9 

17.5 

18.1 

18.8 

19.4 

20.0 

20.7 

21.3 

22.0 

45 

17.3 

17.9 

18.5 

19.2 

19.8 

20.5 

21.2 

21.8 

22.5 

46 

17.7 

18.3 

19.0 

19.6 

20.3 

20.9 

21.6 

22.3 

23.0 

47 

18.1 

18.7 

19.4 

20.0 

20.7 

21.4 

22.1 

22.8 

23.5 

48 

18.4 

19.1 

19.8 

20.5 

21.2 

21.9 

22.6 

23.3 

24.0 

49 

18.8 

19.5 

20.2 

20.9 

21.6 

22.3 

23.0 

23.8 

24.5 

50 

19.2 

19.9 

20.6 

21.3 

22.0 

22.8 

23.5 

24.2 

25.0 

100 

38.4 

39.8 

41.2 

42.6 

44.1 

45.5 

47.0 

48.5 

50.0 

200 

76.9 

79.6 

82.4 

85.3 

88.2 

91.1 

94.0 

97.0 

100.0 

300 

115.3 

119.5 

123.7 

127.9 

132.2 

136.6 

141.0 

145.5 

150.0 

400 

153.7 

159.3 

164.9 

170.6 

176.3 

182.2 

188.1 

194.0 

200.0 

500 

192.2 

199.1 

206.1 

213.2 

220.4 

227.7 

235.0 

242.5 

250.0 

1.62 

1.66 

1.70 

1.74 

1.79 

1.84 

1.89 

1.94 

2.00 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.,  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


Table  2 


177 


To  CHANGE  LONG.  DIFF.  INTO  DEP.  SUBTRACT  TABULAR  NUMBER 
FROM  LONG.  DIFF. 


LONG. 
DIFF. 

MIDDLE  LATITUDE 

OR 

DEP. 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

51 

19.6 

20.3 

21.0 

21.7 

22.5 

23.2 

24.0 

24.7 

25.5 

52 

20.0 

20.7 

21.4 

22.2 

22.9 

23.7 

24.4 

25.2 

26.0 

53 

20.4 

21.1 

21.8 

22.6 

23.4 

24.1 

24.9 

25.7 

26.5 

54 

20.8 

21.5 

22.3 

23.0 

23.8 

24.6 

25.4 

26.2 

27.0 

55 

21.1 

21.9 

22.7 

23.5 

24.2 

25.0 

25.9 

26.7 

27.5 

56 

21.5 

22.3 

23.1 

23.9 

24.7 

25.5 

26.3 

27.2 

28.0 

57 

21.9 

22.7 

23.5 

24.3 

25.1 

26.0 

26.8 

27.6 

28.5 

58 

22.3 

23.1 

23.9 

24.7 

25.6 

26.4 

27.3 

28.1 

29.0 

59 

22.7 

23.5 

24.3 

25.2 

26.0 

26.9 

27.7 

28.6 

29.5 

60 

23.1 

23.9 

24.7 

25.6 

26.4 

27.3 

28.2 

29.1 

30.0 

61 

23.4 

24.3 

25.1 

26.0 

26.9 

27.8 

28.7 

29.6 

30.5 

62 

23.8 

24.7 

25.6 

26.4 

27.3 

28.2 

29.1 

30.1 

31.0 

63 

24.2 

25.1 

26.0 

26.9 

27.8 

28.7 

29.6 

30.6 

31.5 

64 

24.6 

25.5 

26.4 

27.3 

28.2 

29.1 

30.1 

31.0 

32.0 

65 

25.0 

25.9 

26.8 

27.7 

28.7 

29.6 

30.6 

31.5 

32.5 

66 

25.4 

26.3 

27.2 

28.1 

29.1 

30.1 

31.0 

32.0 

33.0 

67 

25.8 

26.7 

27.6 

28.6 

29.5 

30.5 

31.5 

32.5 

33.5 

68 

26.1 

27.1 

28.0 

29.0 

30.0 

31.0 

32.0 

33.0 

34.0 

69 

26.5 

27.5 

28.4 

29.4 

30.4 

31.4 

32.4 

33.5 

34.5 

70 

26.9 

27.9 

28.9 

29.8 

30.9 

31.9 

32.9 

33.9 

35.0 

71 

27.3 

28.3 

29.3 

30.3 

31.3 

32.3 

33.4 

34.4 

35.5 

72 

27.7 

28.7 

29.7 

30.7 

31.7 

32.8 

33.8 

34.9 

36.0 

73 

28.1 

29.1 

30.1 

31.1 

32.2 

33.2 

34.3 

35.4 

36.5 

74 

28.4 

29.5 

30.5 

31.6 

32.6 

33.7 

34.8 

35.9 

37.0 

75 

28.8 

29.9 

30.9 

32.0 

33.1 

34.2 

35.3 

36.4 

37.5 

76 

29.2 

30.3 

31.3 

32.4 

33.5 

34.6 

35.7 

36.9 

38.0 

77 

29.6 

30.7 

31.7 

32.8 

33.9 

35.1 

36.2 

37.3 

38.5 

78 

30.0 

31.1 

32.2 

33.3 

34.4 

35.5 

36.7 

37.8 

39.0 

79 

30.4 

31.5 

32.6 

33.7 

34.8 

36.0 

37.1 

38.3 

39.5 

80 

30.7 

31.9 

33.0 

34.1 

35.3 

36.4 

37.6 

38.8 

40.0 

81 

31.1 

32.3 

33.4 

34.5 

35.7 

36.9 

38.1 

39.3 

40.5 

82 

31.5 

32.7 

33.8 

35.0 

36.1 

37.3 

38.5 

39.8 

41.0 

83 

31.9 

33.0 

34.2 

35.4 

36.6 

37.8 

39.0 

40.3 

41.5 

84 

32.3 

33.4 

34.6 

35.8 

37.0 

38.3 

39.5 

40.7 

42.0 

85 

32.7 

33.8 

35.0 

36.2 

37.5 

38.7 

40.0 

41.2 

42.5 

86 

33.1 

34.2 

35.5 

36.7 

37.9 

39.2 

40.4 

41.7 

43.0 

87 

33.4 

34.6 

35.9 

37.1 

38.4 

39.6 

40.9 

42.2 

43.5 

88 

33.8 

35.0 

36.3 

37.5 

38.8 

40.1 

41.4 

42.7 

44.0 

89 

34.2 

35.4 

36.7 

38.0 

39.2 

40.5 

41.8 

43.2 

44.5 

90 

34.6 

35.8 

37.1 

38.4 

39.7 

41.0 

42.3 

43.6 

45.0 

91 

35.0 

36.2 

37.5 

38.8 

40.1 

41.4 

42.8 

44.1 

45.5 

92 

35.4 

36.6 

37.9 

39.2 

40.6 

41.9 

43.2 

44.6 

46.0 

93 

35.7 

37.0 

38.3 

39.7 

41.0 

42.3 

43.7 

45.1 

46.5 

94 

36.1 

37.4 

38.7 

40.1 

41.4 

42.8 

44.2 

45.6 

47.0 

95 

36.5 

37.8 

39.2 

40.5 

41.9 

43.3 

44.7 

46.1 

47.5 

96 

36.9 

38.2 

39.6 

40.9 

42.3 

43.7 

45.1 

46.6 

48.0 

97 

37.3 

38.6 

40.0 

41.4 

42.8 

44.2 

45.6 

47.0 

48.5 

98 

37.7 

39.0 

40.4 

41.8 

43.2 

44.6 

46.1 

47.5 

49.0 

99 

38.0 

39.4 

40.8 

42.2 

43.6 

45.1 

46.5 

48.0 

49.5 

100 

38.4 

39.8 

41.2 

42.6 

44.1 

45.5 

47.0 

48.5 

50.0 

600 

230.6 

238.9 

247.3 

255.9 

264.5 

273.2 

282.0 

291.0 

300.0 

700 

269.2 

279.7 

288.6 

298.5 

308.6 

318.7 

329.0 

339.6 

350.0 

800 

307.5 

319.5 

329.8 

341.2 

352.6 

364.3 

376.1 

388.0 

400.0 

900 

346.0 

358.3 

371.1 

383.8 

396.8 

409.9 

423.2 

436.6 

450.0 

1.63 

1.66 

1.70 

1.74 

1.79 

1.84 

1.89 

1.94 

2.00 

FACTOR 

To  CHANGE  DEP.  INTO  LONG.  DIFF.  MULTIPLY  TABULAR  NUMBER  BY 
FACTOR  AT  FOOT  OF  COLUMN  AND  ADD  PRODUCT  TO  DEP. 


178 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

100 

00000 

043 

087 

130 

173 

217 

260 

303 

346 

389 

01 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 

44 

43 

42 

02 

860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 

1 

4.4 

4.3 

4.2 

03 

01284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

2 

8.8 

8.6 

8.4 

04 

703 

745 

787 

828 

870 

912 

953 

995 

*036 

*078 

3 

13.2 

12.9 

12.6 

05 

02119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17.6 

17.2 

16.8 

06 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

5 

22.0 

21.5 

21.0 

6 

26.4 

25.8 

25.2 

07 

938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

7 

30.8 

30.1 

29.4 

08 

03342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

8 

35.2 

34.4 

33.6 

09 

743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

9 

39.6 

38.7 

37.8 

110 

04139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

11 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

41 

40 

39 

12 

922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 

1 

4.1 

4.0 

3.9 

13 

05308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

8.2 

8.0 

7.8 

14 
15 
16 

690 
06070 
446 

729 

108 
483 

767 
145 
521 

805 
183 
558 

843 
221 
595 

881 
258 
633 

918 
296 
670 

956 
333 
707 

994 
371 
744 

*032 
408 
781 

3 
4 

5 
6 

12.3 
16.4 
20.5 
24.6 

12.0 
16.0 
20.0 
24.0 

11.7 
15.6 
19.5 
23.4 

17 

819 

856 

893 

930 

967 

*004 

*041 

*078 

*115 

*151 

7 

28.7 

28.0 

27.3 

18 

07188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

8 

32.8 

32.0 

31.2 

19 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

9 

36.9 

36.0 

35.1 

120 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 

21 

08279 

314 

350 

386 

422 

458 

493 

529 

565 

600 

38 

37 

36 

22 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

1 

3.8 

3.7 

3.6 

23 

991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

2 

7.6 

7.4 

7.2 

24 

09342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

3 

11.4 

11.1 

Mo 

10.8 

25 
26 

691 
10037 

726 
072 

760 
106 

795 
140 

830 
175 

864 
209 

899 
243 

934 
278 

968 
312 

*003 
346 

4 
5 
6 

15.2 
19.0 

22.8 

.8 
18.5 
22.2 

14.4 
18.0 
21.6 

27 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

7 

26.6 

25.9 

25.2 

28 

721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

8 

30.4 

29.6 

28.8 

29 

11059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

9 

34.2 

33.3 

32.4 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

31 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 

35 

34 

33 

32 

12057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

1 

3.5 

3.4 

3.3 

33 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

2 

7.0 

6.8 

6.6 

34 
35 
36 

710 
13033 
354 

743 

066 
386 

775 
098 
418 

808 
130 
450 

840 
162 
481 

872 
194 
513 

905 
226 
545 

937 

258 
577 

969 
290 
609 

*001 
322 
640 

3 
4 
5 
6 

10.5 
14.0 
17.5 
21.0 

10.2 
13.6 
17.0 
20.4 

9.9 
13.2 
16.5 
19.8 

37 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

7 

24.5 

23.8 

23.1 

38 

988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

8 

28.0 

27.2 

26.4 

39 

14301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

9 

31.5 

30.6 

29.7 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

41 

922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 

32 

31 

30 

42 

15229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

1 

3.2 

3.1 

3.0 

43 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

2 

6.4 

6.2 

6.0 

44 

836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

3 

19  K 

9.3 
12.4 

9.0 

45 
46 

16  137 
435 

167 
465 

197 
495 

227 
524 

256 
554 

286 
584 

316 
613 

346 
643 

376 
673 

406 
702 

5 
6 

&40»O 

16.0 
19.2 

15^5 
18.6 

15!o 
18.0 

47 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

7 

22.4 

21.7 

21.0 

48 

17026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

8 

25.6 

24.8 

24.0 

49 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

9 

28.8 

27.9 

27.0 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


179 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

150 

17609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

51 

898 

926 

955 

984 

*013 

*041 

*070 

*099 

*127 

*156 

52 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

54 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 

55 

19033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

56 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

58 

866 

893 

921 

948 

976 

*003 

*030 

*058 

«085 

*112 

59 

20140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

61 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

29 

28 

27 

62 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

1 

2.9 

2.8 

2.7 

63 

21219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

5.8 

5.6 

5.4 

64 
65 
66 

484 
748 
22011 

511 

775 
037 

537 
801 
063 

564 
827 
089 

590 
854 
115 

617 
880 
141 

643 
906 
167 

669 
932 
194 

696 
958 
220 

722 
985 
246 

3 
4 
5 
6 

8.7 
11.6 
14.5 
17.4 

8.4 
11.2 
14.0 
16.8 

8.1 
10.8 
13.5 
16.2 

67 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

7 

20.3 

19.6 

18.9 

68 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

8 

23.2 

22.4 

21.6 

69 

789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 

9 

26.1 

25.2 

24.3 

170 

23045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

71 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

26 

25 

24 

72 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

1 

2.6 

2.5 

2.4 

73 

805 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 

2 

5.2 

5.0 

4.8 

74 

24055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

3 

7.8 

7.5 

7.2 

75 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4 

10.4 

10.0 

9.6 

76 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

5 

13.0 

12.5 

12.0 

(5 

15.6 

15.0 

14.4 

77 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 

7 

18.2 

17.5 

16.8 

78 

25042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

8 

20.8 

20.0 

19.2 

79 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

9 

23.4 

22.5 

21.6 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

81 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

23 

22 

21 

82 

26007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 

0  Q 

O  0 

0  1 

83 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

2 

9*0 

4.6 

&*£ 

4.4 

£•4 

4.2 

84 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

6.9 

6.6 

6.3 

85 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9.2 

8.8 

8.4 

86 

951 

975 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 

11.5 

11.0 

10.5 

6 

13.8 

13.2 

12.6 

87 

27184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

7 

16.1 

15.4 

14.7 

88 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

8 

18.4 

17.6 

16.8 

89 

646 

669 

692 

715 

738 

761 

784 

807 

8:30 

852 

9 

20.7 

19.8 

18.9 

190 

875 

898 

921 

944 

967 

989 

*012 

*035 

*058 

*081 

91 

28103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

92 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

93 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

94 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

95 

29003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

96 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

97 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

99 

885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

180 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

01 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

02 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

03 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

04 

963 

984 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

05 

31175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

06 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

07 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

08 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

09 

32015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

Aft 

ft  1 

ft/\ 

11 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

22 

21 

20 

12 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

1 

2.2 

2.1 

2.0 

13 

838 

858 

879 

899 

919 

940 

960 

980 

*001 

*021 

2 

4.4 

4.2 

4.0 

3 

6.6 

6.3 

6.0 

14 

33041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

4 

8.8 

8.4 

8.0 

15 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

5 

11.0 

10.5 

10.0 

16 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

6 

13.2 

12.6 

12.0 

17 
18 
19 

646 
846 
34044 

666 
866 
064 

686 
885 
084 

706 
905 
104 

726 
925 
124 

746 
945 
143 

766 
965 
163 

786 
985 
183 

806 
*005 
203 

826 
*025 
223 

7 
8 
9 

15.4 
17.6 
19.8 

14.7 
16.8 
18.9 

14.0 
16.0 
18.0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

21 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

22 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

23 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*005 

24 

35025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

25 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

26 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

27 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

28 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

29 

984 

*003 

*021 

*040 

*059 

*078 

*097 

*116 

*135 

*154 

230 

36173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

31 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

19 

18 

17 

32 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

1 

1.9 

1.8 

1.7 

33 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

2 

3.8 

3.6 

3.4 

3 

5.7 

5.4 

5.1 

34 

922 

940 

959 

977 

996 

*014 

*033 

*051 

*070 

*088 

4 

y.g 

7.2 

6.8 

35 

37107 

125 

144 

162 

181 

199 

218 

2:36 

254 

273 

5 

95 

9.0 

8*5 

36 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

6 

11.4 

10.8 

10.2 

37 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

7 

13.3 

12.6 

11.9 

38 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

8 

15.2 

14.4 

13.6 

39 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*003 

9 

17.1 

16.2 

15.3 

240 

38021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

41 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

42 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

43 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

44 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

45 

917 

934 

952 

970 

987 

*005 

*023 

*041 

*058 

*076 

46 

39094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

47 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

48 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

49 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts, 

Table  3.    Number  Logarithms 


181 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

250 

39794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

51 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 

52 

40140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

53 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

54 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

55 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

56 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

57 

993 

*010 

*027 

#044 

*061 

*078 

*095 

*111 

*128 

*145 

58 

41162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

59 

330 

347 

3(33 

380 

397 

414 

430 

447 

464 

481 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 

61 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

18   17 

16 

62 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

1   1.8   I/ 

r  1.6 

63 

996 

*012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

*144 

2   3.6   3.^ 

t   3.2 

64 
65 
66 

42160 
325 

488 

177 

341 
504 

193 
357 
521 

210 
374 

537 

226 
390 
553 

243 

406 
570 

259 
423 

586 

275 

439 
602 

292 
455 
619 

308 
472 
635 

3   5.4   5J 
4   7.2   6.* 
5   9.0   8.^ 
(3-  10.8  10.5 

L   4.8 
J   6.4 
>   8.0 
J   9.6 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7  12.6  ll.< 

)  11.2 

68 

813 

830 

846 

8(32 

878 

894 

911 

927 

943 

959 

8  14.4  13.( 

3  12.8 

69 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

9  16.2  15.: 

J  14.4 

270 

43136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

71 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

72 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

73 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

74 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

75 

933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

76 

44091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

77 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

78 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

79 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

81 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 

15 

14 

82 

45025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

11  PC 

1  -I 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

1.0 

2   3.0 

2.S 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3   4.5 

4.2 

85 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

4   6.0 

5.6 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5   7.5 

7.0 

6   9.0 

8.4 

87 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

7  10.5 

9.8 

88 

939 

954 

969 

984 

*000 

*015 

*030 

*045 

*060 

*075 

8  12.0  1 

1.2 

89 

46090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

9  13.5  ] 

2.6 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

91 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

92 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

93 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

94 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

95 

982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

#114 

96 

47129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

97 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

98 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

99 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

182 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

300 

47712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

01 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

02 

48001 

015 

029 

044 

058 

073 

087 

101 

116 

130 

03 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

04 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

05 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

06 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

07 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

08 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

09 

996 

*010 

*024 

*038 

*052 

«066 

*080 

*094 

*108 

*122 

310 

49136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

11 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

15 

14 

12 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 

1   1.5 

1.4 

13 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

2   3.0 

2.8 

3   4.5 

4.2 

14 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

4   6.0 

5.6 

15 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

5   7.5 

7.0 

16 

969 

982 

996 

*010 

*024 

*037 

*051 

*065 

*079 

*092 

6   9X) 

8^4 

17 

50106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

7  10.5 

9.8 

18 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

8  12.0 

11.2 

19 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

9  13.5 

12.6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

21 

651 

664 

678 

691 

705 

718 

732 

745 

ITKQ 

772 

22 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

23 

920 

934 

947 

961 

974 

987 

*001 

*014 

*028 

*041 

24 

51  055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

25 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

26 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

27 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

28 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

29 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

31 

983 

996 

*009 

*022 

*035 

*048 

*061 

*075 

*088 

*101 

13 

12 

32 

52114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

1   1.3 

1.2 

33 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

2   2.6 

2.4 

34 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

3   3.9 
4   5.2 

3.6 

4.8 

35 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

5   6.5 

fi  0 

36 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

6   7^8 

\j,\j 
7.2 

37 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

7   9.1 

8.4 

38 

892 

905 

917 

930 

943 

956 

969 

982 

994 

*007 

8  10.4 

9.6 

39 

53020 

033 

046 

058 

071 

084 

097 

110 

122 

135 

9  11.7 

10.8 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

41 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

42 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

43 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

44 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

45 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

46 

908 

920 

933 

945 

958 

970 

983 

995 

*008 

*020 

47 

54033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

48 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

49 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 

. 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


183 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

350 

54407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

51 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 

52 

(i54 

667 

679 

691 

704 

716 

728 

741 

753 

765 

53 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 

54 

900 

913 

925 

937 

949 

962 

974 

986 

998 

*011 

55 

55023 

035 

047 

060 

072 

084 

096 

108 

121 

133 

56 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

57 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

58 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

59 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

61 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 

13 

12 

62 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

1 

1.3 

1.2 

63 

991 

*003 

*015 

*027 

*038 

*050 

*062 

*074 

*086 

*098 

2 

2.6 

2.4 

64 

56110 

122 

134 

146 

158 

170 

182 

194 

205 

217 

3 

3.9 

3.6 

65 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 

4 

5.2 

4.8 

66 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

5 
6 

6.5 
7.8 

6.0 
7.2 

67 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

7 

9.1 

8.4 

68 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

8 

10.4 

9.6 

69 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

9 

11.7 

10.8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

71 

937 

949 

961 

972 

984 

996 

*008 

*019 

*031 

*043 

72 

57054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

73 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

74 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

75 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 

76 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 

77 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 

78 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

79 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

380 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

81 

58092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

11 

10 

82 

206 

213 

229 

240 

252 

263 

274 

286 

297 

309 

1 

I  "I 

1  0 

83 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

m 

2 

J..X 

2.2 

J.-VJ 

2.0 

84 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

3 

3.3 

3.0 

85 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

4 

4.4 

4.0 

86 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

5 

5.5 

5.0 

6 

6.6 

6.0 

87 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

7 

7.7 

7.0 

88 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 

8 

8.8 

8.0 

89 

995 

*006 

*017 

*028 

*040 

*051 

*062 

*073 

*084 

*095 

9 

9.9 

9.0 

390 

59106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

91 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

92 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

93 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

94 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 

95 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

96 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

97 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

98 

988 

999 

*010 

*021 

*032 

*04.". 

*054 

*065 

*076 

*086 

99 

60097 

108 

119 

130 

141 

152 

163 

173 

184 

195 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

184 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

400 

60206 

217 

228 

23!) 

249 

260 

271 

282 

293 

304 

01 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

02 

423 

433 

444 

455 

4(5(5 

477 

487 

498 

509 

520 

03 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

04 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

05 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

06 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

07 

959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 

06 

61  066 

077 

087 

0!)8 

109 

119 

130 

140 

151 

162 

09 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

11 

384 

395 

405 

416 

426 

437 

448 

458 

4(59 

479 

12 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

13 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

14 

700 

711 

721 

731 

742 

752 

'763 

773 

784 

794 

15 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

16 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 

17 

62014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

18 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 

19 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

21 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 

11  10  9 

22 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

1  1.1  1.0  0.9 

23 

634 

644 

655 

(565 

675 

685 

696 

706 

716 

726 

2  2.2  2.0  l!8 

24 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3  3.3  3.0  2.7 

25 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

4  4.4  4.0  3.6 

26 

941 

951 

961 

972 

982 

992 

*002 

*012 

*022 

*033 

5  5.5  5.0  4.5 

6  6.6  6.0  5.4 

27 

63043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

7  7.7  7.0  6.3 

28 

144 

155 

1(55 

175 

185 

195 

205 

215 

225 

23(5 

8  8.8  8.0  7.2 

29 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

9  9.9  9.0  8.1 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

31 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

32 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

33 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

34 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

35 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

36 

949 

959 

969 

979 

988 

998 

*008 

*018 

*028 

*038 

37 

64048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

38 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

39 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

41 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

42 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

43 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

44 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

45 

836 

846 

856 

866 

875 

885 

895 

904 

914 

924 

46 

933 

943 

953 

963 

972 

982 

992 

*002 

*011 

*021 

47 

65031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

48 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

49 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3.    Number  Logarithms 


185 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

450 

65  321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

51 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

52 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

53 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

54 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

55 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

56 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

57 

992 

*001 

*011 

*020 

*030 

*039 

*049 

*058 

*068 

*077 

58 

66087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

59 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

61 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

82 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

63 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

64 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

65 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

66 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

67 

932 

941 

950 

960 

969 

978 

987 

997 

*006 

*015 

68 

67025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

69 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

71 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 

10  9   8 

72 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

1  1.0  0.9  0.8 

73 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

2  2.0  1.8  1.6 

74 
75 
76 

578 
669 
761 

587 
679 
770 

596 
688 
779 

605 

697 
788 

614 
706 

797 

624 
715 

806 

633 

724 
815 

642 
733 

825 

651 
742 

834 

660 
752 
843 

3  3.0  2.7  2.4 
4  4.0  3.6  3.2 
5  5.0  4.5  4.0 
6  6.0  5.4  4.8 

77 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

7  7.0  6.3  5.6 

78 

943 

952 

961 

970 

979 

988 

997 

*006 

*015 

*024 

8  8.0  7.2  6.4 

79 

68034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

9  9.0  8.1  7.2 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

81 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

82 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

83 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

84 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

85 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

86 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

87 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

88 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

89 

931 

940 

949 

958 

9(J6 

975 

984 

993 

*002 

*011 

490 

69020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

91 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

92 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

93 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

94 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

95 

461 

469 

478 

487 

496 

504 

513 

•522 

531 

539 

96 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

97 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 

98 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

99 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

0 

1  i  2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

186 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

01 

984 

992 

*001 

*010 

*018 

*027 

*036 

*044 

*053 

*062 

02 

70070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

03 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

04 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

05 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

06 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

07 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

08 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

09 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

11 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

12 

927 

935 

944 

952 

961 

9(59 

978 

986 

995 

*003 

13 

71012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

14 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

15 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

16 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

17 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 

18 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

19 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

21 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 

987 

22 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

1  0.9  0.8  0.7 

23 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2  1.8  1.6  1.4 

24 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

3  2.7  2.4  2.1 

25 

72016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

4  3.6  3.2  2.8 

26 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5  4.5  4.0  3.5 

6  5.4  4.8  4.2 

27 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

7  6.3  5.6  4.9 

28 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

8  7.2  6.4  5.6 

29 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

9  8.1  7.2  6.3 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

31 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 

32 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 

33 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

34 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 

35 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

36 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

37 

997 

*006 

*014 

*022 

*030 

*038 

*046 

*054 

*062 

*070 

38 

73078 

086 

094 

102 

111 

119 

127 

135 

143 

151 

39 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

41 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

42 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

43 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

44 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

45 

640 

648 

656 

664 

672 

•679 

687 

695 

703 

711 

46 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

47 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

48 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 

49 

957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


187 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts, 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

51 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 

52 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

53 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

54 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

55 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

56 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 

57 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

58 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

59 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

I 

61 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 

62 

974 

981 

989 

997 

*005 

*012 

*020 

*028 

*035 

*043 

63 

75051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

64 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

65 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

66 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

67 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

68 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

69 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

71 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

8   7 

72 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

1   0.8   0.7 

73 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

2   1.6   1.4 

74 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

3   2.4   2.1 

75 

967 

974 

982 

989 

997 

*005 

*012 

*020 

*027 

*035 

4   3.2   2.8 

76 

76042 

050 

057 

065 

072 

080 

087 

095 

103 

110 

5   4.0   3.5 

6   4.8   4.2 

77 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 

7   5.6   4.9 

78 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

8   6.4   5.6 

79 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 

9   7.2   6.3 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

81 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

82 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

83 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

84 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

85 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

86 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

87 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

88 

938 

945 

953 

960 

967 

975 

982 

989 

997 

*004 

89 

77012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

91 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

92 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

93 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 

94 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

95 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

96 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 

97 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

98 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

99 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

188 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

600 

77815 

822 

8:30 

837 

844 

851 

859 

866 

873 

880 

01 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

02 

960 

967 

974 

981 

988 

996 

*003 

*010 

*017 

*025 

03 

78032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

04 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 

05 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

06 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

07 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

08 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

09 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

11 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

12 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

13 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

14 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

15 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 

16 

958 

965 

972 

979 

986 

993 

*000 

*007 

*014 

*021 

17 

79029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

18 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

19 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

21 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 

876 

22 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

1  0.8  0.7  0.6 

23 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

2  1.6  1.4  1.2 

24 
25 
26 

518 
588 
657 

525 
595 
664 

532 
602 
671 

539 
609 
678 

546 

616 
685 

553 
623 
692 

560 
630 
699 

567 
637 
706 

574 
644 
713 

581 
650 
720 

3  2.4  2.1  1.8 
4  3.2  2.8  2.4 
5  4.0  3.5  3.0 
6  4.8  4.2  3.6 

27 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

7  5.6  4.9  4.2 

28 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

8  6.4  5.6  4.8 

29 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

9  7.2  6.3  5.4 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

31 

80003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

32 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

33 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

34 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

35 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

36 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

37 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

38 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

39 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

41 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

42 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

43 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

44 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

45 

956 

963 

969 

976 

983 

990 

99(5 

*003 

*010 

•017 

46 

81023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

47 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 

48 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

49 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts, 

Table  3.    Number  Logarithms 


189 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts. 

650 

81291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

51 

358 

365 

371 

378 

385 

391. 

398 

405 

411 

418 

52 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

53 

491 

498 

605 

511 

518 

525 

531 

538 

544 

551 

54 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

55 

624 

631 

637 

(544 

651 

657 

6(54 

671 

677 

684 

56 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

57 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

58 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

59 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 

61 

82020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

62 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

63 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

64 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

65 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

66 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

67 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

68 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

69 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

71 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

7    6 

72 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

1   0.7   0.6 

73 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

2   1.4   1.2 

74 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

3   2.1   1.8 

75 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

4   2.8   2.4 

76 

995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 

5   3.5   3.0 
6   4.2   3.6 

77 

83059 

065 

072 

078 

085 

091 

097 

104 

110 

117 

7   4.9   4.2 

78 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

8   5.6   4.8 

79 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

9   6.3   5.4 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

81 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

82 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

83 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

84 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

85 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

86 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

87 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

88 

759 

705 

771 

778 

784 

790 

797 

803 

809 

816 

89 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

91 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 

92 

84011 

017 

023 

029 

036 

042 

048 

055 

061 

067 

93 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 

94 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

95 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

96 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

97 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

98 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

99 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

190 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts. 

700 

84510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

01 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

02 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

03 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

04 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

05 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

06 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

07 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

08 

85003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

09 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

11 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

12 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

13 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

14 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

15 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

16 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

17 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

18 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

19 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

21 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

765 

22 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

1  0.7  0.6  0.5 

23 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

2  1.4  1.2  1.0 

24 
25 
26 

974 
86034 
094 

980 
040 
100 

986 
046 
106 

992 
052 
112 

998 
058 
118 

*004 
064 
124 

*010 
070 
130 

*016 
076 
136 

*022 
082 
141 

*028 
088 
147 

3  2.1  1.8  1.5 
4  2.8  2.4  2.0 
5  3.5  3.0  2.5 

6  4.2  3.6  3.0 

27 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

7  4.9  4.2  3.5 

28 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

8  5.6  4.8  4.0 

29 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9  6.3  5.4  4.5 

730 

332 

338 

344 

350 

356 

362 

3(58 

374 

380 

386 

31 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

32 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

33 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

34 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

35 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

36 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

37 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

38 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

39 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

41 

982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

42 

87040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

43 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

44 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

45 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

46 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

47 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

48 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

49 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


191 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

750 

87506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

51 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

52 

622 

628 

633 

639 

645 

651 

656 

662 

6(58 

674 

53 

<>7'.» 

685 

691 

697 

703 

708 

714 

720 

726 

731 

54 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

55 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

56 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

57 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

58 

967 

973 

978 

984 

990 

996 

*001 

*007 

*013 

*018 

59 

88024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

61 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

62 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

63 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

64 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

65 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

66 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

67 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

68 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

69 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

71 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 

6    5 

72 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

1   0.6   0.5 

73 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

2   1.2   1.0 

74 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

3   1.8   1.5 

75 

930 

936 

941 

947 

953 

958 

9(54 

969 

975 

981 

4   2.4   2.0 

76 

986 

992 

997 

*003 

*009 

*014 

*020 

*025 

*031 

*037 

5   3.0   2.5 

6   3.6   3.0 

77 

89042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

7   4.2   3.5 

78 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 

8   4.8   4.0 

79 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 

9   5.4   4.5 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

81 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 

82 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

83 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

84 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

85 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

86 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

87 

597 

603 

609 

614 

(520 

625 

631 

636 

642 

647 

88 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

89 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

91 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

92 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 

93 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

&4 

982 

988 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 

95 

90037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

96 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

97 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

98 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

99 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pta, 

192 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

800 

90309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

6   5 

1   0.6   0.5 
2   1.2   1.0 
3   1.8   1.5 
4   2.4   2.0 
5   3.0   2.5 
6   3.6   3.0 
7   4.2   3.5 
8   4.8   4.0 
9   5.4   4.5 

01 
02 
03 

04 
05 
06 

07 

08 
09 

363 
417 
472 

526 
580 
634 

687 
741 
795 

369 
423 

477 

531 
585 
639 

693 

747 
800 

374 

428 
482 

536 
590 
644 

698 
752 
806 

380 
434 
488 

542 
596 
650 

703 

757 
811 

385 
439 
493 

547 
601 
655 

709 
763 
816 

390 
445 
499 

553 
607 
660 

714 
768 
822 

396 
450 
504 

558 
612 
666 

720 
773 

827 

401 
455 
509 

563 
617 
671 

725 
779 
832 

407 
461 
515 

569 
623 

677 

730 

784 
838 

412 
466 
520 

574 

628 
682 

736 

789 
843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

11 
12 
13 

14 
15 
16 

17 
18 
19 

902 
956 
91009 

062 
116 
169 

222 
275 

328 

907 
961 
014 

068 
121 
174 

228 
281 
334 

913 
966 
020 

073 
126 
180 

233 

286 
339 

918 
972 
025 

078 
132 
185 

238 
291 
344 

924 

977 
030 

084 
137 
190 

243 

297 
350 

929 
982 
036 

089 
142 
196 

249 
302 
355 

934 
988 
041 

094 
148 
201 

254 

307 
360 

940 
993 
046 

100 
153 
206 

259 
312 

365 

945 
998 
052 

105 
158 
212 

265 
318 
371 

950 
*004 
057 

110 
164 
217 

270 
323 
376 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

21 
22 
23 

24 
25 
26 

27 
28 
29 

434 
487 
540 

593 
645 

698 

751 
803 
855 

440 
492 
545 

598 
651 
703 

756 
808 
861 

445 
498 
551 

603 
656 
709 

761 

814 
866 

450 
503 
556 

609 
661 
714 

766 
819 
871 

455 
508 
561 

614 

666 
719 

772 
824 
876 

461 
514 
566 

619 
672 
724 

777 
829 
882 

466 
519 

572 

624 

677 
730 

782 
834 
887 

471 
524 

577 

630 
682 
735 

787 
840 
892 

477 
529 
582 

635 

687 
740 

793 

845 
897 

482 
535 
587 

640 
693 
745 

798 
850 
903 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

31 
32 
33 

34 

35 
36 

37 
38 
39 

960 
92012 
065 

117 
169 
221 

273 
324 
376 

965 
018 
070 

122 
174 
226 

278 
330 
381 

971 
023 
075 

127 
179 
231 

283 
335 

387 

976 

028 
080 

132 
184 
236 

288 
340 
392 

981 
033 

085 

137 
189 
241 

293 
345 

397 

986 
038 
091 

143 

195 

247 

298 
350 
402 

991 
044 
096 

148 
200 
252 

304 
355 

407 

997 
049 
101 

153 

205 
257 

309 
361 
412 

*002 
054 
106 

158 
210 

262 

314 

366 
418 

*007 
059 
111 

163 
215 

267 

319 
371 
423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

41 
42 
43 

44 
45 
46 

47 
48 
49 

480 
531 
583 

634 

686 
737 

788 
840 
891 

485 
536 
588 

639 
691 
742 

793 

845 
896 

490 
542 
593 

645 
696 

747 

799 
850 
901 

495 
547 
598 

650 
701 

752 

804 
855 
906 

500 
552 
603 

655 
706 
758 

809 
860 
911 

505 
557 
609 

660 
711 
763 

814 
865 
916 

511 
562 
614 

665 
716 

768 

819 
870 
921 

516 
567 
619 

670 
722 
773 

824 
875 
927 

521 
572 
624 

675 

727 
778 

829 
881 
932 

526 
578 
629 

681 
732 
783 

834 
886 
937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

0 

1 

g 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


193 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

51 

993 

998 

*003 

*008 

*013 

*018 

*024 

*029 

*034 

«039 

52 

93044 

049 

054 

059 

064 

069 

075 

080 

085 

090 

53 

095 

100 

105 

110 

115- 

120 

125 

131 

136 

141 

54 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 

55 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

56 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

57 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

58 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

59 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

860 

4,50 

455 

460 

465 

470 

475 

480 

485 

490 

495 

61 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

(52 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

63 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

64 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

65 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

66 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

67 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

68 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

69 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

71 

94002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

654 

72 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

1  0.6  0.5  0.4 

73 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

2  1.2  1.0  0.8 

74 
75 
76 

151 
201 

250 

156 
206 
255 

161 

211 
260 

166 
216 
265 

171 
221 

270 

176 

226 
275 

181 
231 
280 

186 
236 

285 

191 

240 
290 

196 
245 
295 

3  1.8  1.5  1.2 
4  2.4  2.0  1.6 
5  3.0  2.5  2.0 
6  3.6  3.0  2.4 

77 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

7  4.2  3.5  2.8 

78 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

8  4.8  4.0  3,2 

79 

395) 

404 

409 

414 

419 

424 

429 

433 

438 

443 

9  5.4  4.5  3.6 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

81 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

82 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

83 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

84 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

85 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

86 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

87 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

88 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

89 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

890 

9o9 

944 

949 

954 

959 

963 

968 

973 

978 

983 

91 

988 

993 

998 

*002 

*007 

*012 

*017 

*022 

«027 

*032 

92 

95  Otfi 

041 

046 

051 

056 

061 

066 

071 

075 

080 

93 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

94 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

95 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

96 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

97 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

98 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

99 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

194 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts= 

900 

95424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

01 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

02 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

03 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

04 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

05 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

06 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

07 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

08 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

09 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

11 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

12 

999 

«004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 

13 

96.047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

14 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

15 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

16 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

17 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

18 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

19 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

21 

426 

431 

4a5 

440 

445 

450 

454 

459 

464 

468 

5   4 

22 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

1   0.5   0.4 

23 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

2   1.0   0.8 

24 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

3   1.5   1.2 

25 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

4   2.0   1.6 

26 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

5   2.5   2.0 

6   3.0   2.4 

27 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

7   3.5   2.8 

28 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

8   4.0   3.2 

29 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

9   4.5   3.6 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

31 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

32 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

33 

988 

993 

997 

*002 

*007 

*011 

*016 

*021 

*025 

*030 

34 

97035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

35 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

36 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

37 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

38 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

39 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

41 

359 

3T4 

368 

373 

377 

382 

387 

391 

396 

400 

42 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

43 

451 

.456 

460 

465 

470 

474 

479 

483 

488 

493 

44 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

45 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

46 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

47 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

48 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

49 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

Table  3.    Number  Logarithms 


195 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

950 

97772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

51 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

52 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

53 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

54 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

55 

98000 

005 

009 

014 

019 

023 

028 

032 

037 

041 

56 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

57 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

58 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

59 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

61 

272 

277 

281 

286 

290 

295 

299 

304 

303 

313 

62 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

63 

363 

3G7 

372 

376 

381 

385 

390 

394 

399 

403 

64 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

65 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

66 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

67 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

68 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

69 

632 

(>37 

641 

646 

650 

655 

659 

664 

668 

673 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

71 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

5   4 

72 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

1  0.5  0.4 

73 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

2  1.0  0.8 

74 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

3  1.5  1.2 

75 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

4  2.Q  1.6 

76 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

5  2.5  2.0 

6  3.0  2.4 

77 

989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 

7  3.5  2.8 

78 

99034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

8  4.0  3.2 

79 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

9  4.5  3.6 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

81 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

82 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

83 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

84 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

85 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

86 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

87 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

88 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

89 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

91 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

92 

651 

656 

660 

664 

669 

673 

677 

082 

686 

691 

93 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

94 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

95 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

96 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

97 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

98 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

99 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

1000 

00000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts, 

196 


Table  4.    Trigonometric  Logarithms 


0°  (180°) 


(359°)  179° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

— 

0.00  000 





0.00  000 



60 

1 

6.46  373 

.00  000 

6.46  373 

3.53  627 

.00  000 

3.53  627 

59 

2 

6.76  476 

.00  000 

6.76  476 

3.23  524 

.00  000 

.23  524 

58 

3 

6.94  085 

.00  000 

6.94  085 

3.05  915 

.00  000 

.05  915 

57 

4 

7.06  579 

.00  000 

7.06  579 

2.93  421 

.00  000 

2.93  421 

56 

5 

7.16  270 

0.00  000 

7.16  270 

2.83  730 

0.00  000 

2.83  730 

55 

6 

.24  188 

.00  000 

.24  188 

.75  812 

.00  000 

.75  812 

54 

7 

.30  882 

.00  000 

.30  882 

.69  118 

.00  000 

.69  118 

53 

8 

.36  682 

.00  000 

.36  682 

.63  318 

.00  000 

.63  318 

52 

9 

.41  797 

.00  000 

.41  797 

.58  203 

.00  000 

.58  203 

51 

10 

7.46  373 

0.00  000 

7.46  373 

2.53  627 

0.00  000 

2.53  627 

50 

11 

.50  512 

.00  000 

.50  512 

.49  488 

.00  000 

.49  488 

49 

12 

.54  291 

.00  000 

.54  291 

.45  709 

.00  000 

.45  709 

48 

13 

.57  767 

.00  000 

.57  767 

.42  233 

.00  000 

.42  233 

47 

14 

.60  985 

.00  000 

.60  986 

.39  014 

.00  000 

.39  015 

46 

15 

7.63  982 

0.00  000 

7.63  982 

2.36  018 

0.00  000 

2.36  018 

45 

16 

.66  784 

.00  000 

.66  785 

.33  215 

.00  000 

.33  216 

44 

17 

.69417 

9.99  999 

.69  418 

.30  582 

.00  001 

.30583 

43 

18 

.71  900 

.99  999 

.71  900 

.28  100 

.00  001 

.28  100 

42 

19 

.74  248 

.99  999 

.74  248 

.25  752 

.00  001 

.25  752 

41 

20 

7.76  475 

9.99  999 

7.76  476 

2.23  524 

0.00  001 

2.23  525 

40 

21 

.78  594 

.99  999 

.78  595 

.21  405 

.00  001 

.21  406 

39 

22 

.80  615 

.99  999 

.80  615 

.19  385 

.00  001 

.19  385 

38 

23 

.82  545 

.99  999 

.82  546 

.17454 

.00  001 

.17  455 

37 

24 

.84  393 

.99  999 

.84  394 

.15  606 

.00  001 

.15  607 

36 

25 

7.86  166 

9.99  999 

7.86  167 

2.13  833 

0.00  001 

2.13  834 

35 

26 

.87  870 

.99  999 

.87  871 

.12  129 

.00  001 

.12  130 

34 

27 

.89  509 

.99  999 

.89  510 

.10  490 

.00  001 

.10491 

33 

28 

.91  088 

.99  999 

.91  089 

.08911 

.00  001 

.08  912 

32 

29 

.92  612 

.99  998 

.92  613 

.07  387 

.00  002 

.07  388 

31 

30 

7.94  084 

9.99  998 

7.94  086 

2.05  914 

0.00  002 

2.05  916 

30 

31 

.95  508 

.99  998 

.95  510 

.04  490 

.00  002 

.04  492 

29 

32 

.96  887 

.99  998 

.96  889 

.03  111 

.00  002 

.03  113 

28 

33 

.98  223 

.99  998 

.98  225 

.01  775 

.00  002 

.01  777 

27 

34 

.99  520 

.99  998 

.99  522 

.00  478 

.00  002 

.00  480 

26 

35 

8.00  779 

9.99  998 

8.00  781 

1.99219 

0.00  002 

1.99  221 

25 

36 

.02  002 

.99  998 

.02  004 

.97  996 

.00  002 

.97  998 

24 

37 

.03  192 

.99  997 

.03  194 

.96  806 

.00  003 

.96  808 

23 

38 

.04  350 

.99  997 

.  .04  353 

.95  647 

.00  003 

.95  650 

22 

39 

.05  478 

.99  997 

.05  481 

.94  519 

.00  003 

.94  522 

21 

40 

8.06  578 

9.99  997 

8.06  581 

1.93419 

0.00  003 

1.93422 

20 

41 

.07  650 

.99  997 

.07  653 

.92  347 

.00  003 

.92  350 

19 

42 

.08  696 

.99  997 

.08  700 

.91  300 

.00  003 

.91  304 

18 

43 

.09  718 

.99  997 

.09  722 

.90  278 

.00  003 

.90  282 

17 

44 

.10717 

.99  996 

.10  720 

.89  280 

.00  004 

.89  283 

16 

45 

8.11  693 

9.99  996 

8.11  696 

1.88304 

0.00  004 

1.88307 

15 

46 

.12  647 

.99  996 

.12651 

.87  349 

.00  004 

.87  353 

14 

47 

.13  581 

.99  996 

.13  585 

.86  415 

.00  004 

.86  419 

13 

48 

.14  495 

.99  996 

.14  500 

.85  500 

.00  004 

.85  505 

12 

49 

.15391 

.99  996 

.15  395 

.84605 

.00  004 

.84  609 

11 

50 

8.16268 

9.99  995 

8.16  273 

1.83727 

0.00  005 

1.83  732 

10 

51 

.17  128 

.99  995 

.17  133 

.82  867 

.00  005 

.82  872 

9 

52 

.17971 

.99  995 

.17  976 

.82  024 

.00  005 

.82  029 

8 

53 

.18  798 

.99  995 

.18804 

.81  196 

.00  005 

.81  202 

7 

54 

.19610 

.99  995 

.19616 

.80  384 

.00  005 

.80  390 

6 

55 

8.20  407 

9.99  994 

8.20413 

1.79  587 

0.00  006 

1.79  593 

5 

56 

.21  189 

.99  994 

.21  195 

.78  805 

.00  006 

.78811 

4 

57 

.21  958 

.99  994 

.21  964 

.78  036 

.00  006 

.78  042 

3 

58 

.22  713 

.99  994 

.22  720 

.77  280 

.00  006 

.77  287 

2 

59 

.23  456 

.99  994 

.23  462 

.76  538 

.00  006 

.76  544 

1 

60 

8.24  186 

9.99  993 

8.24  192 

1.75808 

0.00  007 

1.75  814 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

1* 

90°  (270°) 


(269°)  89° 


Table  4.    Trigonometric  Logarithms 


197 


1°  (181°) 


(358°)  178° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.24  186 

9.99  993 

8.24  192 

1.75  808 

0.00  007 

1.75  814 

60 

1 

.24  903 

.99  993 

.24  910 

.75  090 

.00  007 

.75  097 

59 

2 

.25  609 

.99  993 

.25  616 

.74384 

.00  007 

.74  391 

58 

3 

.26  304 

.99  993 

.26  312 

.73  688 

.00  007 

.73  696 

57 

4 

.26  988 

.99  992 

.26  996 

.73  004 

.00  008 

.73  012 

56 

5 

8.27  661 

9.99  992 

8.27  669 

1.72  331 

0.00  008 

1.72  339 

55 

6 

.28  324 

.99  992 

.28  332 

.71  668 

.00  008 

.71  676 

54 

7 

.28  977 

.99  992 

.28  986 

.71  014 

.00  008 

.71  023 

53 

8 

.29  621 

.99  992 

.29  629 

.70  371 

.00  008 

.70  379 

52 

9 

.30  255 

.99  991 

.30  263 

.69  737 

.00  009 

.69  745 

51 

10 

8.30  879 

9.99  991 

8.30  888 

1.69  112 

0.00  009 

1.69  121 

50 

11 

.31  495 

.99  991 

.31  505 

.68  495 

.00  009 

.68  505 

49 

12 

.32  103 

.99  990 

.32  112 

.67  888 

.00  010 

.67  897 

48 

13 

.32  702 

.99  990 

.32711 

.67  289 

.00  010 

.67  298 

47 

14 

.33  292 

.99  990 

.33  302 

.66  698 

.00  010 

.66  708 

46 

15 

8.33  875 

9.99  990 

8.33  886 

1.66  114 

0.00  010 

1.66  125 

45 

16 

.34  450 

.99  989 

.34  461 

.65  539 

00.011 

65550 

44 

17 

.35  018 

.99  989 

.35  029 

.64  971 

.00011 

.64  982 

43 

18 

.35  578 

.99  989 

.35  590 

.64  410 

.00011 

.64  422 

42 

19 

.36  131 

.99  989 

.36  143 

.63  857 

.00011 

.63  869 

41 

20 

8.36  678 

9.99  988 

8.36  689 

1.63311 

0.00  012 

1.63  322 

40 

21 

.37  217 

.99  988 

.37  229 

.62  771 

.00  012 

.62  783 

39 

22 

.37  750 

.99  988 

.37  762 

.62  238 

.00  012 

.62  250 

38 

23 

.38  276 

.99  987 

.38  289 

.61  711 

.00  013 

.61  724 

37 

24 

.38  796 

.99  987 

.38  809 

.61  191 

.00  013 

.61  204 

36 

25 

8.39  310 

9.99  987 

8.39  323 

1.60677 

0.00  013 

1.60  690 

35 

26 

.39  818 

.99  986 

.39  832 

.60  168 

.00  014 

.60  182 

34 

27 

.40  320 

.99  986 

.40  334 

.59  666 

.00  014 

.59  680 

33 

28 

.40  816 

.99  986 

.40  830 

.59  170 

.00  014 

.59  184 

32 

29 

.41  307 

.99  985 

.41  321 

.58  679 

.00  015 

.58  693 

31 

30 

8.41  792 

9.99  985 

8.41  807 

1.58  193 

0.00  015 

1.58208 

30 

31 

.42  272 

.99  985 

.42  287 

.57713 

.00  015 

.57  728 

29 

32 

.42  746 

.99  984 

.42  762 

.57  238 

.00  016 

.57  254 

28 

33 

.43  216 

.99984 

.43  232 

.56  768 

.00  016 

.56  784 

27 

34 

.43  680 

.99984 

.43  696 

.56  304 

.00  016 

.56  320 

26 

35 

8.44  139 

9.99  983 

8.44  156 

1.55  844 

0.00  017 

1.55  861 

25 

36 

.44  594 

.99  983 

.44611 

.55  389 

.00  017 

.55  406 

24 

37 

.45  044 

.99  983 

.45  061 

.54  939 

.00  017 

.54  956 

23 

38 

.45  489 

.99  982 

.45  507 

.54  493 

.00  018 

.54  511 

22 

39 

.45  930 

.99  982 

.45  948 

.54  052 

.00  018 

.54  070 

21 

40 

8.46  366 

9.99  982 

8.46  385 

1.53615 

0.00  018 

1.53  634 

20 

41 

.46  799 

.99  981 

.46  817 

.53  183 

.00  019 

.53  201 

19 

42 

.47  226 

.99  981 

.47  245 

.52755 

.00  019 

.52  774 

18 

43 

.47  650 

.99  981 

.47  669 

.52  331 

.00  019 

.52  350 

17 

44 

.48  069 

.99  980 

.48  089 

.51  911 

.00  020 

.51  931 

16 

45 

8.48  485 

9.99  980 

8.48  505 

1.51  495 

0.00  020 

1.51  515 

15 

46 

.48  896 

.99  979 

.48  917 

.51  083 

.00  021 

.51  104 

14 

47 

.49  304 

.99  979 

.49  325 

.50  675 

.00  021 

.50  696 

13 

48 

.49  708 

.99  979 

.49  729 

.50  271 

.00  021 

.50  292 

12 

49 

.50  108 

.99  978 

.50  130 

.49  870 

.00  022 

.49  892 

11 

50 

8.50  504 

9.99  978 

8.50  527 

1.49  473 

0.00  022 

1.49  496 

10 

51 

.50  897 

.99  977 

.50  920 

.49  080 

.00  023 

.49  103 

9 

52 

.51  287 

.99  977 

.51  310 

.48  690 

.00  023 

.48  713 

8 

53 

.51  673 

.99  977 

.51  696 

.48  304 

.00  023 

.48  327 

7 

54 

.52  055 

.99  976 

.52  079 

.47  921 

.00  024 

.47  945 

6 

55 

8.52  434 

9.99  976 

8.52  459 

1.47  541 

0.00  024 

1.47  566 

5 

56 

.52  810 

.99  975 

.52  835 

.47  165 

.00  025 

.47  190 

4 

57 

.53  183 

.99  975 

.53  208 

.46  792 

.00  025 

.46  817 

3 

58 

.53  552 

.99  974 

.53  578 

.46  422 

.00  026 

.46  448 

2 

59 

.53  919 

.99  974 

.53  945 

.46  055 

.00  026 

.46  081 

1 

60 

8.54  282 

9.99  974 

8.54  308 

1.45692 

0.00  026 

1.45718 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

91°  (271°) 


(268°)  88° 


198 


Table  4.    Trigonometric  Logarithms 


2°  (182°) 


(357°)  177C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.54  282 

9.99  974 

8.54  308 

1.45  692 

0.00  026 

1.45  718 

60 

1 

.54  642 

.99  973 

.54  669 

.45  331 

.00  027 

.45  358 

59 

2 

.54  999 

.99  973 

.55  027 

.44  973 

.00  027 

.45  001 

58 

3 

.55  354 

.99  972 

.55  382 

.44  618 

.00  028 

.44  646 

57 

4 

.55  705 

.99  972 

.55  734 

.44  266 

.00  028 

.44  295 

56 

5 

8.56  054 

9.99  971 

8.56  083 

1.43  917 

0.00  029 

1.43  946 

55 

6 

.56  400 

.99  971 

.56  429 

.43  571 

.00  029 

.43  600 

54 

7 

.56  743 

.99  970 

.56  773 

.43  227 

.00  030 

.43  257 

53 

8 

.57  084 

.99  970 

.57  114 

.42  886 

.00  030 

.42  916 

52 

9 

.57  421 

.99  969 

.57  452 

.42  548 

.00  031 

.42  579 

51 

10 

8.57  757 

9.99  969 

8.57  788 

1.42212 

0.00  031 

1.42  243 

50 

11 

.58  089 

.99  968 

.58  121 

.41  879 

.00  032 

.41  911 

49 

12 

.58  419 

.99  968 

.58  451 

.41  549 

.00  032 

.41  581 

48 

13 

.58  747 

.99  967 

.58  779 

.41  221 

.00  033 

.41  253 

47 

14 

.59  072 

.99  967 

.59  105 

.40895 

.00  033 

.40  928 

46 

15 

8.59  395 

9.99  967 

8.59  428 

1.40572 

0.00  033 

1.40  605 

45 

16 

.59  715 

.99  966 

.59  749 

.40  251 

.00  034 

.40  285 

44 

17 

.60  033 

.99  966 

.60  068 

.39  932 

.00  034 

.39  967 

43 

18 

.60  349 

.99  965 

.60  384 

.39  616 

.00  035 

.39  651 

42 

19 

.60  662 

.99  964 

.60  698 

.39  302 

.00  036 

.39  338 

41 

20 

8.60  973 

9.99  964 

8.61  009 

1.38991 

0.00  036 

1.39  027 

40 

21 

.61  282 

.99  963 

.61  319 

.38  681 

.00  037 

.38  718 

39 

22 

.61  589 

.99  963 

.61  626 

.38  374 

.00  037 

.38411 

38 

23 

.61  894 

.99  962 

.61  931 

.38  069 

.00  038 

.38  106 

37 

24 

.62  196 

.99  962 

.62  234 

.37  766 

.00  038 

.37  804 

36 

25 

8.62  497 

9.99  961 

8.62  535 

1.37465 

0.00  039 

1.37503 

35 

26 

.62  795 

.99  961 

.62  834 

.37  166 

.00  039 

.37  205 

34 

27 

.63  091 

.99  960 

.63  131 

.36  869 

.00  940 

.36  909 

33 

28 

.63  385 

.99  960 

.63  426 

.36  574 

.00  040 

.36  615 

32 

29 

.63  678 

.99  959 

.63  718 

.36  282 

.00  041 

.36  322 

31 

30 

8.63  968 

9.99  959 

8.64  009 

1.35  991 

0.00  041 

1.36032 

30 

31 

.64  256 

.99  958 

.64  298 

.35  702 

.00  042 

.35  744 

29 

32 

.64  543 

.99  958 

.64  585 

.35415 

.00  042 

.35  457 

28 

33 

.64  827 

.99  957 

.64  870 

.35  130 

.00  043 

.35  173 

27 

34 

.65  110 

.99  956 

.65  154 

.34846 

.00  044 

.34  890 

26 

35 

8.65  391 

9.99  956 

8.65  435 

1.34  565 

0.00  044 

1.34  609 

25 

36 

.65  670 

.99  955 

.65  715 

.34  285 

.00  045 

.34  330 

24 

37 

.65  947 

.99  955 

.65  993 

.34  007 

.00  045 

.34  053 

23 

38 

.66  223 

.99  954 

.66  269 

.33  731 

.00  046 

.33  777 

22 

39 

.66  497 

.99  954 

.66  543 

.33  457 

.00  046 

.33  503 

21 

40 

8.66  769 

9.99  953 

8.66  816 

1.33  184 

0.00  047 

1.33231 

20 

41 

.67  039 

.99  952 

.67  087 

.32  913 

.00  048 

.32  961 

19 

42 

.67  308 

.99  952 

.67  356 

.32  644 

.00  048 

.32  692 

18 

43 

.67  575 

.99  951 

.67  624 

.32  376 

.00  049 

.32  425 

17 

44 

.67  841 

.99  951 

.67  890 

.32  110 

.00  049 

.32  159 

16 

45 

8.68  104 

9.99  950 

8.68  154 

1.31  846 

0.00  050 

1.31  896 

15 

46 

.68  367 

.99  949 

.68417 

.31  583 

.00  051 

.31  633 

14 

47 

.68  627 

.99  949 

.68  678 

.31  322 

.00  051 

.31  373 

13 

48 

.68  886 

.99  948 

.68  938 

.31  062 

.00  052 

.31  114 

12 

49 

.69  144 

.99  948 

.69  196 

.30  804 

.00  052 

.30  856 

11 

50 

8.69  400 

9.99  947 

8.69  453 

1.30  547 

0.00  053 

1.30  600 

10 

51 

.69  654 

.99  946 

.69  708 

.30  292 

.00  054 

.30  346 

9 

52 

.69  907 

.99  946 

.69  962 

.30  038 

.00  054 

.30  093 

8 

53 

.70  159 

.99  945 

.70  214 

.29  786 

.00  055 

.29  841 

7 

54 

.70  409 

.99  944 

.70  465 

.29  535 

.00  056 

.29  591 

6 

55 

8.70  658 

9.99  944 

8.70  714 

1.29  286 

0.00  056 

1.29  342 

5 

56 

.70  905 

.99  943 

.70  962 

.29  038 

.00  057 

.29  095 

4 

57 

.71  151 

.99  942 

.71  208 

.28  792 

.00  058 

.28849 

3 

58 

.71  395 

.99  942 

.71  453 

.28  547 

.00  058 

.28  605 

2 

59 

.71  638 

.99  941 

.71  697 

.28  303 

.00  059 

.28  362 

1 

60 

8.71  880 

9.99  940 

8.71  940 

1.28060 

0.00  060 

1.28  120 

0 

Cos 

Sin 

Cot   1   Tan 

Csc 

Sec 

' 

92°  (272°) 


(267°)  87° 


Table  4.    Trigonometric  Logarithms 


199 


3°  (183°) 


(356°)  176° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.71  880 

9.99  940 

8.71  940 

1.28  060 

0.00  060 

1.28  120 

60 

1 

.72  120 

.99  940 

.72  181 

.27  819 

.00  060 

.27  880 

59 

2 

.72  359 

.99  939 

.72  420 

.27  580 

.00  061 

.27  641 

58 

3 

.72  597 

.99  938 

.72  659 

.27  341 

.00  062 

.27  403 

57 

4 

.72  834 

.99  938 

.72  896 

.27  104 

.00  062 

.27  166 

56 

5 

8.73  069 

9.99  937 

8.73  132 

1.26  868 

0.00  063 

1.26931 

55 

6 

.73  303 

.99  936 

.73  366 

.26  634 

.00  064 

.26  697 

54 

7 

.73  535 

.99  936 

.73  600 

.26  400 

.00  064 

.26  465 

53 

8 

.73  767 

.99  935 

.73832 

.26  168 

.00  065 

.26  233 

52 

9 

.73  997 

.99  934 

.74  063 

.25  937 

.00  066 

.26  003 

51 

10 

8.74  226 

9.99  934 

8.74  292 

1.25  708 

0.00  066 

1.25774 

50 

11 

.74  454 

.99  933 

.74  521 

.25  479 

.00  067 

.25  546 

49 

12 

.74  680 

.99  932 

.74  748 

.25  252 

.00  068 

.25  320 

48 

13 

.74  906 

.99  932 

.74  974 

.25  026 

.00  068 

.25  094 

47 

14 

.75  130 

.99  931 

.75  199 

.24  801 

.00  069 

.24  870 

46 

15 

8.75  353 

9.99  930 

8.75  423 

1.24577 

0.00  070 

1.24  647 

45 

16 

.75  575 

.99  929 

.75  645 

.24  355 

.00  071 

.24  425 

44 

17 

.75  795 

.99  929 

.75  867 

.24  133 

.00  071 

.24  205 

43 

18 

.76  015 

.99  928 

.76  087 

.23  913 

.00  072 

.23  985 

42 

19 

.76  234 

.99  927 

.76  306 

.23  694 

.00  073 

.23  766 

41 

20 

8.76  451 

9.99  926 

8.76  525 

1.23  475 

0.00  074 

1.23  549 

40 

21 

.76  667 

.99  926 

.76  742 

.23  258 

.00  074 

.23  333 

39 

22 

.76  883 

.99  925 

.76  958 

.23  042 

.00  075 

.23  117 

38 

23 

.77  097 

.99  924 

.77  173 

.22  827 

.00  076 

.22  903 

37 

24 

.77  310 

.99  923 

.77  387 

.22  613 

.00  077 

.22  690 

36 

25 

8.77  522 

9.99  923 

8.77  600 

1.22  400 

0.00  077 

1.22  478 

35 

26 

.77  733 

.99  922 

.77811 

.22  189 

.00  078 

.22  267 

34 

27 

.77  943 

.99  921 

.78  022 

.21  978 

.00  079 

.22  057 

33 

28 

.78  152 

.99  920 

.78  232 

.21  768 

.00  080 

.21  848 

32 

29 

.78  360 

.99  920 

.78  441 

.21  559 

.00  080 

.21  640 

31 

30 

8.78  568 

9.99  919 

8.78  649 

1.21  351 

0.00  081 

1.21  432 

30 

31 

.78  774 

.99  918 

.78855 

.21  145 

.00  082 

.21  226 

29 

32 

.78  979 

.99  917 

.79  061 

.20  939 

.00  083 

.21  021 

28 

33 

.79  183 

.99  917 

.79  266 

.20  734 

.00  083 

.20  817 

27 

34 

.79  386 

.99  916 

.79  470 

.20  530 

.00084 

.20  614 

26 

35 

8.79  588 

9.99  915 

8.79  673 

1.20  327 

0.00  085 

1.20412 

25 

36 

.79  789 

.99  914 

.79  875 

.20  125 

.00  086 

.20211 

24 

37 

.79  990 

.99  913 

.80  076 

.19324 

.00  087 

.20  010 

23 

38 

.80  189 

.99  913 

.80  277 

.19  723 

.00  087 

.19811 

22 

39 

.80  388 

.99  912 

.80  476 

.19  524 

.00  088 

.19612 

21 

40 

8.80  585 

9.99911 

8.80  674 

1.19  326 

0.00  089 

1.19415 

20 

41 

.80  782 

.99  910 

.80  872 

.19  128 

.00  090 

.19218 

19 

42 

.80  978 

.99  909 

.81  068 

.18  932 

.00  091 

.19  022 

18 

43 

.81  173 

.99  909 

.81  264 

.18  736 

.00  091 

.18  827 

17 

44 

.81  367 

.99  908 

.81  459 

.18541 

.00  092 

.18  633 

16 

45 

8.81  560 

9.99  907 

8.81  653 

1.18347 

0.00  093 

1.18440 

15 

46 

.81  752 

.99  906 

.81  846 

.18  154 

.00  094 

.18  248 

14 

47 

.81  944 

.99  905 

.82  038 

.17  962 

.00  095 

.18  056 

13 

48 

.82  134 

.99  904 

.82  230 

.17  770 

.00  096 

.17  866 

12 

49 

.82  324 

.99  904 

.82  420 

.17  580 

.00  096 

.17676 

11 

50 

8.82  513 

9.99  903 

8.82  610 

1.17  390 

0.00  097 

1.17  487 

10 

51 

.82  701 

.99  902 

.82  799 

.17  201 

.00  098 

.17  299 

9 

52 

.82  888 

.99  901 

.82  987 

.17013 

.00  099 

.17112 

8 

53 

.83  075 

.99  900 

.83  175 

.16  825 

.00  100 

.16  925 

7 

54 

.83  261 

.99  899 

.83361 

.16  639 

.00  101 

.16  739 

6 

55 

8.83  446 

9.99  898 

8.83  547 

1.16453 

0.00  102 

1.16554 

5 

56 

.83  630 

.99  898 

.83  732 

.16  268 

.00  102 

.16  370 

4 

57 

.83  813 

.99  897 

.83916 

.16084 

.00  103 

.16  187 

3 

58 

.83  996 

.99  896 

.84  100 

.15  900 

.00  104 

.16  004 

2 

59 

.84177 

.99  895 

.84282 

.15718 

.00  105 

.15  823 

1 

60 

8.84  358 

9.99  894 

8.84  464 

1.15536 

0.00  106 

1.15  642 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

93°  (273°) 


(266°)  86C 


200 


Table  4:.    Trigonometric  Logarithms 


4°  (184°) 


(355°)  175° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.84  358 

9.99  894 

8.84  464 

1.15  536 

0.00  106 

1.15  642 

60 

1 

.84  539 

.99  893 

.84646 

.15  354 

.00  107 

.15461 

59 

2 

.84718 

.99  892 

.84826 

.15  174 

.00  108 

.15  282 

58 

3 

.84  897 

.99  891 

.85006 

.14  994 

.00  109 

.15  103 

57 

4 

.85  075 

.99  891 

.85  185 

.14  815 

.00  109 

.14  925 

56 

5 

8.85  252 

9.99  890 

8.85  363 

1.14  637 

0.00  110 

1.14  748 

55 

6 

.85  429 

.99  889 

.85  540 

.14  460 

.00  111 

.14  571 

54 

7 

.85  605 

.99  888 

.85  717 

.14  283 

.00  112 

.14  395 

53 

8 

.85  780 

.99  887 

.85  893 

.14  107 

.00  113 

.14  220 

52 

9 

.85  955 

.99  886 

.86  069 

.13  931 

.00  114 

.14  045 

51 

10 

8.86  128 

9.99  885 

8.86  243 

1.13  757 

0.00  115 

1.13  872 

50 

11 

.86  301 

.99884 

.86  417 

.13  583 

.00  116 

.13  699 

49 

12 

.86  474 

.99  883 

.86  591 

.13409 

.00  117 

.13  526 

48 

13 

.86  645 

.99  882 

.86  763 

.13  237 

.00  118 

.13  355 

47 

14 

.86  816 

.99  881 

.86  935 

.13  065 

.00  119 

.13  184 

46 

15 

8.86  987 

9.99  880 

8.87  106 

1.12  894 

0.00  120 

1.13013 

45 

16 

.87  156 

.99  879 

.87  277 

.12  723 

.00  121 

.12844 

'44 

17 

.87  325 

.99  879 

.87  447 

.12  553 

.00  121 

.12  675 

43 

18 

.87  494 

.99  878 

.87  616 

.12384 

.00  122 

.12506 

42 

19 

.87  661 

.99  877 

.87  785 

.12215 

.00  123 

.12  339 

41 

20 

8.87  829 

9.99  876 

8.87  953 

1.12  047 

0.00  124 

1.12  171 

40 

21 

.87  995 

.99  875 

.88  120 

.11  880 

.00  125 

.12  005 

39 

22 

.88  161 

.99  874 

.88  287 

.11713 

00  126 

.11  839 

38 

23 

.88  326 

.99  873 

.88  453 

.11  547 

.00  127 

.11  674 

37 

24 

.88  490 

.99  872 

.88  618 

.11  382 

.00  128 

.11  510 

36 

25 

8.88  654 

9.99  871 

8.88  783 

1.11  217 

0.00  129 

1.11  346 

35 

26 

.88  817 

.99  870 

.88  948 

.11  052 

.00  130 

.11  183 

34 

27 

.88  980 

.99  869 

.89  111 

.10  889 

.00131 

.11  020 

33 

28 

.89  142 

.99  868 

.89  274 

.10  726 

.00  132 

.10  858 

32 

29 

.89  304 

.99  867 

.89  437 

.10563 

.00  133 

.10  696 

31 

30 

8.89  464 

9.99  866 

8.89  598 

1.10402 

0.00  134 

1.10536 

30 

31 

.89  625 

.99  865 

.89  760 

.10  240 

.00  135 

.10  375 

29 

32 

.89784 

.99  864 

.89  920 

.10  080 

.00  136 

.10216 

28 

33 

.89  943 

.99  863 

.90  080 

.09  920 

.00  137 

.10  057 

27 

34 

.90  102 

.99  862 

.90  240 

.09  760 

.00  138 

.09  898 

26 

35 

8.90  260 

9.99  861 

8.90  399 

1.09  601 

0.00  139 

1.09  740 

25 

36 

.90417 

.99  860 

.90  557 

.09  443 

.00  140 

.09  583 

24 

37 

.90  574 

.99  859 

.90  715 

.09  285 

.00  141 

.09  426 

23 

38 

.90  730 

.99  858 

.90  872 

.09  128 

.00  142 

.09  270 

22 

39 

.90  885 

.99  857 

.91  029 

.08  971 

.00  143 

.09  115 

21 

40 

8.91  040 

9.99  856 

8.91  185 

1.08815 

0.00  144 

1.08  960 

20 

41 

.91  195 

.99  855 

.91  340 

.08  660 

.00  145 

.08  805 

19 

42 

.91  349 

.99  854 

.91  495 

.08  505 

.00  146 

.08  651 

18 

43 

.91  502 

.99  853 

.91  650 

.08  350 

.00  147 

.08  498 

17 

44 

.91  655 

.99  852 

.91  803 

.08  197 

.00  148 

.08  345 

16 

45 

8.91  807 

9.99  851 

8.91  957 

1.08043 

0.00  149 

1.08  193 

15 

46 

.91  959 

.99  850 

.92  110 

.07  890 

.00  150 

.08  041 

14 

47 

.92  110 

.99  848 

.92  262 

.07  738 

.00  152 

.07  890 

13 

48 

.92  261 

.99  847 

.92  414 

.07  586 

.00  153 

.07  739 

12 

49 

.92411 

.99846 

.92  565 

.07  435 

.00  154 

.07  589 

11 

50 

8.92  561 

9.99  845 

8.92  716 

1.07  284 

0.00  155 

1.07  439 

10 

51 

.92  710 

.99  844 

.92  866 

.07  134 

.00  156 

.07  290 

9 

52 

.92  859 

.99  843 

.93  016 

.06984 

.00  157 

.07  141 

8 

53 

.93  007 

.99  842 

.93  165 

.06  835 

.00  158 

.06  993 

7 

54 

.93  154 

.99  841 

.93  313 

.06  687 

.00  159 

.06  846 

6 

55 

8.93  301 

9.99  840 

8.93  462 

1.06  538 

0.00  160 

1.06  699 

5 

56 

.93  448 

.99  839 

.93  609 

.06  391 

.00  161 

.06  552 

4 

57 

.93  594 

.99  838 

.93  756 

.06  244 

.00  162 

.06  406 

3 

58 

.93  740 

.99  837 

.93  903 

.06  097 

.00  163 

.06  260 

2 

59 

.93  885 

.99  836 

.94  049 

.05  951 

.00  164 

.06  115 

1 

60 

8.94  030 

9.99  834 

8.94  195 

1.05  805 

0.00  166 

1.05  970 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

94°  (274°) 


(265°)  85° 


Table  4.    Trigonometric  Logarithms 


201 


5°  (185°) 


(354°)  174° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.94  030 

9.99  834 

8.94  195 

l.UO  &U£> 

0.00  166 

1.05  970 

60 

1 

.94  174 

.99  833 

.94  340 

.05  660 

.00  167 

.05  826 

59 

2 

.94  317 

.99  832 

.94  485 

.05  515 

.00  168 

.05  683 

58 

3 

.94  461 

.99  831 

.94  630 

.05  370 

.00169 

.05  539 

57 

4 

.94  603 

.99  830 

.94  773 

.05227 

.00  170 

.05  397 

56 

5 

8.94  746 

9.99  829 

8.94  917 

1.05083 

0.00  171 

1.05  254 

55 

6 

.94  887 

.99  828 

.95  060 

.04  940 

.00  172 

.05  113 

54 

7 

.95  029 

.99  827 

.95  202 

.04798 

.00  173 

.04  971 

53 

8 

.95  170 

.99  825 

.95344 

.04  656 

.00  175 

.04  830 

52 

9 

.95  310 

.99  824 

.95  486 

.04  514 

.00  176 

.04  690 

51 

10 

8.95  450 

9.99  823 

8.95  627 

1.04  373 

0.00  177 

1.04  550 

50 

11 

.95  589 

.99  822 

.95  767 

.04233 

.00  178 

.04411 

49 

12 

.95  728 

.99  821 

.95  908 

.04  092 

.00  179 

.04  272 

48 

13 

.95  867 

.99  820 

.96  047 

.03  953 

.00  180 

.04  133 

47 

14 

.96  005 

.99  819 

.96  187 

.03  813 

.00181 

.03  995 

46 

15 

8.96  143 

9.99  817 

8.96  325 

1.03  675 

0.00  183 

1.03  857 

45 

16 

.96  280 

.99  816 

.96  464 

.03  536 

.00  184 

.03  720 

44 

17 

.96  417 

.99  815 

.96  602 

.03  398 

.00  185 

.03  583 

43 

18 

.96  553 

.99  814 

.96  739 

.03  261 

.00  186 

.03  447 

42 

19 

.96  689 

.99  813 

.96  877 

.03  123 

.00  187 

.03311 

41 

20 

8.96  825 

9.99  812 

8.97  013 

1.02987 

0.00  188 

1.03  175 

40 

21 

.96  960 

.99  810 

.97  150 

.02  850 

.00  190 

.03  040 

39 

22 

.97  095 

.99  809 

.97  285 

.02  715 

.00  191 

.02  905 

38 

23 

.97  229 

.99  808 

.97  421 

.02  579 

.00  192 

.02  771 

37 

24 

.97  363 

.99  807 

.97  556 

.02  444 

.00  193 

.02  637 

36 

25 

8.97  496 

9.99  806 

8.97  691 

1.02309 

0.00  194 

1.02  504 

35 

26 

.97  629 

.99  804 

.97  825 

.02  175 

.00  196 

.02  371 

34 

27 

.97  762 

.99  803 

.97  959 

.02  041 

.00  197 

.02  238 

33 

28 

.97  894 

.99  802 

.98  092 

.01  908 

.00  198 

.02  106 

32 

29 

.98  026 

.99  801 

.98  225 

.01  775 

.00  199 

.01  974 

31 

30 

8.98  157 

9.99  800 

8.98  358 

1.01  642 

0.00  200 

1.01  843 

30 

31 

.98  288 

.99  798 

.98  490 

.01  510 

.00  202 

.01  712 

29 

32 

.98419 

.99  797 

.98  622 

.01  378 

.00  203 

.01  581 

28 

33 

.98549 

.99  796 

.98  753 

.01  247 

.00  204 

.01  451 

27 

34 

.98  679 

.99  795 

.98884 

.01  116 

.00  205 

.01  321 

26 

35 

8.98  808 

9.99  793 

8.99  015 

1.00985 

0.00  207 

1.01  192 

25 

36 

.98  937 

.99  792 

.99  145 

.00  855 

.00  208 

.01  063 

24 

37 

.99066 

.99  791 

.99  275 

.00  725 

.00  209 

.00934 

23 

38 

.99  194 

.99  790 

.99  405 

.00  595 

.00  210 

.00  806 

22 

39 

.99  322 

.99  788 

.99  534 

.00  466 

.00  212 

.00678 

21 

40 

8.99  450 

9.99  787 

8.99  662 

1.00338 

0.00  213 

1.00550 

20 

41 

.99  577 

.99  786 

.99  791 

.00  209 

.00  214 

.00  423 

19 

42 

.99  704 

.99  785 

.99  919 

.00  081 

.00215 

.00  296 

18 

43 

.99  830 

.99  783 

9.00  046 

0.99  954 

.00217 

.00  170 

17 

44 

.99  956 

.99  782 

.00  174 

.99  826 

.00  218 

.00  044 

16 

45 

9.00  082 

9.99  781 

9.00  301 

0.99  699 

0.00  219 

0.99  918 

15 

46 

.00207 

.99  780 

.00  427 

.99  573 

.00  220 

.99  793 

14 

47 

.00  332 

.99  778 

.00  553 

.99  447 

.00  222 

.99  668 

13 

48 

.00  456 

.99  777 

.00  679 

.99  321 

.00  223 

.99  544 

12 

49 

.00  581 

.99  776 

.00  805 

.99  195 

.00  224 

.99419 

11 

50 

9.00  704 

9.99  775 

9.00  930 

0.99  070 

0.00  225 

0.99  296 

10 

51 

.00828 

.99  773 

.01  055 

.98  945 

.00  227 

.99  172 

9 

52 

.00  951 

.99  772 

.01  179 

.98  821 

.00  228 

.99  049 

8 

53 

.01  074 

.99  771 

.01  303 

.98  697 

.00  229 

.98  926 

7 

54 

.01  196 

.99  769 

.01  427 

.98  573 

.00231 

.98  804 

6 

55 

9.01  318 

9.99  768 

9.01  550 

0.98  450 

0.00  232 

0.98  682 

5 

56 

.01  440 

.99  767 

.01  673 

.98  327 

.00  233 

.98  560 

4 

57 

.01  561 

.99  765 

.01  796 

.98  204 

.00  235 

.98  439 

3 

58 

.01  682 

.99764 

.01  918 

.98  082 

.00  236 

.98318 

2 

59 

.01  803 

.99  763 

.02040 

.97  960 

.00  237 

.98  197 

1 

60 

9.01  923 

9.99  761 

9.02  162 

0.97  838 

0.00  239 

0.98  077 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

95°  (275°) 


(264°)  84° 


202 


Table  4.    Trigonometric  Logarithms 


6°  (186°) 


(353°)  173C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.01  923 

9.99  761 

9.02  162 

0.97  838 

0.00  239 

0.98  077 

60 

1 

.02  043 

.99  760 

.02  283 

.97  717 

00240 

.97  957 

59 

2 

.02  163 

.99  759 

.02  404 

.97  596 

.00  241 

.97  837 

58 

3 

.02  283 

.99  757 

.02  525 

.97  475 

.00  243 

.97  717 

57 

4 

.02  402 

.99  756 

.02  645 

.97  355 

.00  244 

.97  598 

56 

5 

9.02  520 

9.99  755 

9.02  766 

0.97  234 

0.00  245 

0.97  480 

55 

6 

.02  639 

.99  753 

.02  885 

.97  115 

.00  247 

.97  361 

54 

7 

.02  757 

.99  752 

.03  005 

.96  995 

.00  248 

.97  243 

53 

8 

.02  874 

.99  751 

.03  124 

.96  876 

.00  249 

.97  126 

52 

9 

.02  992 

.99  749 

.03  242 

.96  758 

.00  251 

.97  008 

51 

10 

9.03  109 

9.99  748 

9.03  361 

0.96  639 

0.00  252 

0.96  891 

50 

11 

.03  226 

.99  747 

.03  479 

.96  521 

.00  253 

.96  774 

49 

12 

.03  342 

.99  745 

.03  597 

.96  403 

.00  255 

.96  658 

48 

13 

.03  458 

.99  744 

.03  714 

.96  286 

.00  256 

.96  542 

47 

14 

.03  574 

.99  742 

.03  832 

.96  168 

.00  258 

.96  426 

46 

15 

9.03  690 

9.99  741 

9.03  948 

0.96  052 

0.00  259 

0.96  310 

45 

16 

.03  805 

.99  740 

.04  065 

.95  935 

.00  260 

.96  195 

44 

17 

.03  920 

.99  738 

.04  181 

.95  819 

.00  262 

.96  080 

43 

18 

.04  034 

.99  737 

.04  297 

.95  703 

.00  263 

.95  966 

42 

19 

.04  149 

.99  736 

.04  413 

.95  587 

.00  264 

.95  851 

41 

20 

9.04  262 

9.99  734 

9.04  528 

0.95  472 

0.00  266 

0.95  738 

40 

21 

.04  376 

.99  733 

.04  643 

.95  357 

.00  267 

.95  624 

39 

22 

.04  490 

.99  731 

.04  758 

.95  242 

.00  269 

.95  510 

38 

23 

.04  603 

.99  730 

.04  873 

.95  127 

.00  270 

.95  397 

37 

24 

.04  715 

.99  728 

.04  987 

.95  013 

.00  272 

.95  285 

36 

25 

9.04  828 

9.99  727 

9.05  101 

0.94  899 

0.00  273 

0.95  172 

35 

26 

.04  940 

.99  726 

.05  214 

.94  786 

.00  274 

.95  060 

34 

27 

.05  052 

.99  724 

.05  328 

.94  672 

.00  276 

.94  948 

33 

28 

.05  164 

.99  723 

.05  441 

.  94559 

.00  277 

.94  836 

32 

29 

.05  275 

.99  721 

.05  553 

.94  447 

.00  279 

.94  725 

31 

30 

9.05  386 

9.99  720 

9.05  666 

0.94  334 

0.00  280 

0.94  614 

30 

31 

.05  497 

.99  718 

.05  778 

.94  222 

.00  282 

.94  503 

29 

32 

.05  607 

.99  717 

.05  890 

.94  110 

.00  283 

.94  393 

28 

33 

.05  717 

.99  716 

.06  002 

.93  998 

.00  284 

.94  283 

27 

34 

.05  827 

.99  714 

.06  113 

.93  887 

.00  286 

.94  173 

26 

35 

9.05  937 

9.99  713 

9.06  224 

0.93  776 

0.00  287 

0.94  063 

25 

36 

.06  046 

.99  711 

.06  335 

.93  665 

.00  289 

.93  954 

24 

37 

.06  155 

.99  710 

.06  445 

.93555 

.00  290 

.93  845 

23 

38 

.06  264 

.99  708 

.06  556 

.93  444 

.00  292 

.93  736 

22 

39 

.06  372 

.99  707 

.06  666 

.93  334 

.00  293 

.93  628 

21 

40 

9.06  481 

9.99  705 

9.06  775 

0.93  225 

0.00  295 

0.93  519 

20 

41 

.06  589 

.99  704 

.06  885 

.93  115 

.00  296 

.93411 

19 

42 

.06  696 

.99  702 

.06  994 

.93  006 

.00  298 

.93  304 

18 

43 

.06  804 

.99  701 

.07  103 

.92  897 

.00  299 

.93  196 

17 

44 

.06911 

.99  699 

.07211 

.92  789 

.00  301 

.93  089 

16 

45 

9.07  018 

9.99  698 

9.07  320 

0.92  680 

0.00  302 

0.92  982 

15 

46 

.07  124 

.99  696 

.07  428 

.92  572 

.00  304 

.92  876 

14 

47 

.07  231 

.99  695 

.07  536 

.92  464 

.00  305 

.92  769 

13 

48 

.07  337 

.99  693 

.07  643 

.92  357 

00.307 

.92  663 

12 

49 

.07  442 

.99  692 

.07  751 

.92  249 

.00  308 

.92  558 

11 

50 

9.07  548 

9.99  690 

9.07  858 

0.92  142 

0.00  310 

0.92  452 

10 

51 

.07  653 

.99  689 

.07  964 

.92  036 

.00311 

.92  347 

9 

52 

.07  758 

.99  687 

.08  071 

.91  929 

.00  313 

.92  242 

8 

53 

.07  863 

.99  686 

.08  177 

.91  823 

.00  314 

.92  137 

7 

54 

.07  968 

.99  684 

.08  283 

.91  717 

.00316 

.92  032 

6 

55 

9.08  072 

9.99  683 

9.08  389 

0.91  611 

0.00317 

0.91  928 

5 

56 

.08  176 

.99  681 

.08  495 

.91  505 

.00  319 

.91  824 

4 

57 

.08  280 

.99  680 

.08  600 

.91  400 

.00  320 

.91  720 

3 

58 

.08  383 

.99  678 

.08  705 

.91  295 

.00  322 

.91  617 

2 

59 

.08  486 

.99  677 

.08  810 

.91  190 

.00  323 

.91  514 

1 

60 

9.08  589 

9.99  675 

9.08  914 

0.91  086 

0.00  325 

0.91411 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

(276°) 


(263°)  83° 


Table  4.    Trigonometric  Logarithms 


203 


7°  (187°) 


(352°)  172C 


/ 

Sin 

Cos 

Tan 

Cot 

Sec      Csc 

0 

9.08  589 

9.99  675 

9.08  914 

0.91  086 

0.00  325 

0.91  411 

60 

1 

.08  692 

.99  674 

.09  019 

.90  981 

.00326 

.91  308 

59 

2 

.08  795 

.99  672 

.09  123 

.90  877 

.00328 

.91  205 

58 

3 

.08  897 

.99  670 

.09  227 

.90  773 

.00  330 

.91  103 

57 

4 

.08  999 

.99  669 

.09  330 

.90  670 

.00331 

.91  001 

56 

5 

9.09  101 

9.99  667 

9.09  434 

0.90  566 

0.00  333 

0.90  899 

55 

6 

.09  202 

.99  666 

.09  537 

.90  463 

.00334 

.90  798 

54 

7 

.09  304 

.99  664 

.09  640 

.90  360 

.00  336 

.90  696 

53 

8 

.09  405 

.99  663 

.09  742 

.90  258 

.00  337 

.90  595 

52 

9 

.09  506 

.99  661 

.09845 

.90  155 

.00339 

.90  494 

51 

10 

9.09  606 

9.99  659 

9.09  947 

0.90  053 

0.00  34  1 

0.90  394 

50 

11 

.09  707 

.99  658 

.10  049 

.89  951 

.00  342 

.90  293 

49 

12 

.09  807 

.99  656 

.10  150 

.89  850 

.00344 

.90  193 

48 

13 

.09  907 

.99  655 

.10  252 

.89  748 

.00  345 

.90  093 

47 

14 

.10  006 

.99  653 

.10  353 

.89647 

.00347 

.89  994 

46 

15 

9.10  106 

9.99  651 

9.10  454 

0.89  546 

0.00  349 

0.89  894 

45 

16 

.10  205 

.99  650 

.10  555 

.89  445 

.00  350 

.89  795 

44 

17 

.10304 

.99648 

.10  656 

.89  344 

.00352 

.89  696 

43 

18 

.10  402 

.99  647 

.10  756 

.89  244 

.00  353 

.89  598 

42 

19 

.10  501 

.99645 

.10  856 

.89  144 

.00  355 

.89  499 

41 

20 

'9.  10  599 

9.99  643 

9.10  956 

0.89  044 

0.00  357 

0.89  401 

40 

21 

.10  697 

.99642 

.11  056 

.88  944 

.00  358 

.89  303 

39 

22 

.10  795 

.99640 

.11  155 

.88845 

.00  360 

.89  205 

38 

23 

.10  893 

.99  638 

.11  254 

.88  746 

.00  362 

.89  107 

37 

24 

.10  990 

.99  637 

.11  353 

.88647 

.00  363 

.89  010 

36 

25 

9.11  087 

9.99  635 

9.11  452 

0.88  548 

0.00  365 

0.88  913 

35 

26 

.11  184 

.99  633 

.11551 

.88  449 

.00  367 

.88  816 

34 

27 

.11  281 

.99  632 

.11649 

.88  351 

.00  368 

.88  719 

33 

28 

.11  377 

.99  630 

.11  747 

.88  253 

.00  370 

.88  623 

32 

29 

.11  474 

.99  629 

.11  845 

.88  155 

.00  371 

.88  526 

31 

30 

9.11  570 

9.99  627 

9.11  943 

0.88  057 

0.00  373 

0.88  430 

30 

31 

.11  666 

.99  625 

.12  040 

.87  960 

.00375 

.88  334 

29 

32 

.1.1  761 

.99  624 

.12  138 

.87  862 

.00  376 

.88  239 

28 

33 

.11  857 

.99  622 

.12  235 

.87  765 

.00  378 

.88  143 

27 

34 

.11  952 

.99  620 

.12  332 

.87  668 

.00  380 

.88048 

26 

35 

9.12  047 

9.99  618 

9.12  428 

0.87  572 

0.00  382 

0.87  953 

25 

36 

.12  142 

.99  617 

.12  525 

.87  475 

.00383 

.87  858 

24 

37 

.12  236 

.99  615 

.12  621 

.87  379 

.00  385 

.87  764 

23 

38 

.12331 

.99  613 

.12717 

.87  283 

.00  387 

.87  669 

22 

39 

.12  425 

.99  612 

.12813 

.87  187 

.00388 

.87  575 

21 

40 

9.12519 

9.99  610 

9.12  909 

0.87  091 

0.00  390 

0.87  481 

20 

41 

.12612 

.99  608 

.13004 

.86  996 

.00  392 

.87  388 

19 

42 

.12  706 

.99  607 

.13  099 

.86  901 

.00  393 

.87  294 

18 

43 

.12  799 

.99  605 

.13  194 

.86  806 

.00  395 

.87  201 

17 

44 

.12  892 

.99  603 

.13  289 

.86711 

.00  397 

.87  108 

16 

45 

9.12  985 

9.99  601 

9.13  384 

0.86  616 

0.00  399 

0.87  015 

15 

46 

.13  078 

.99  600 

.13  478 

.86  522 

.00  400 

.86  922 

14 

47 

.13  171 

.99  598 

.13  573 

.86  427 

.00402 

.86  829 

13 

48 

.13  263 

.99  596 

.13  667 

.86  333 

.00  404 

.86  737 

12 

49 

.13  355 

.99  595 

.13  761 

.86  239 

.00405 

.86  645 

11 

50 

9.13  447 

9.99  593 

9.13  854 

0.86  146 

0.00  407 

0.86  553 

10 

51 

.13  539 

.99  591 

.13  948 

.86  052 

.00  409 

.86  461 

9 

52 

.13  630 

.99589 

.14041 

.85  959 

.00411 

.86  370 

8 

53 

.13  722 

.99  588 

.14  134 

.85  866 

.00412 

.86  278 

7 

54 

.13  813 

.99  586 

.14  227 

.85  773 

.00  414 

.86  187 

6 

55 

9.13  904 

9.99  584 

9.14  320 

0.85  680 

0.00416 

0.86  096 

5 

56 

.13  994 

.99  582 

.14412 

.85  588 

.00  418 

.86  006 

4 

57 

.14  085 

.99  581 

.14  504 

.85  496 

.00  419 

.85  915 

3 

58 

.14  175 

.99  579 

.14  597 

.85  403 

.00  421 

.85825 

2 

59 

.14  266 

.99  577 

.14  688 

.85  312 

.00423 

.85  734 

1 

60 

9.14  356 

9.99  575 

9.14  780 

0.85  220 

0.00  425 

0.85  644 

0 

Cos 

Sin 

Cot 

Tan 

Csc      Sec 

' 

97°  (277°) 


(262°)  82{ 


204 


Table  4.    Trigonometric  Logarithms 


8°  (188°) 


(351°)  171C 


Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.14  356 

9.99  575 

9.14  780 

0.85  220 

0.00  425 

0.85  644 

60 

1 

.14445 

.99  574 

.14872 

.85  128 

.00  426 

.85  555 

59 

2 

.14  535 

.99  572 

.14  963 

.65  037 

.00  428 

.85  465 

58 

3 

.14  624 

.99  570 

.15  054 

.84  946 

.00  430 

.85  376 

57 

4 

.14  714 

.99  568 

.15  145 

.84  855 

.00  432 

.85  286 

56 

5 

9.14  803 

9.99  566 

9.15  236 

0.84  764 

0.00  434 

0.85  197 

55 

6 

.14  891 

.99  565 

.15327 

.84  673 

.00  435 

.85  109 

54 

7 

.14  980 

.99  563 

.15417 

.84  583 

.00  437 

.85  020 

53 

8 

.15069 

.99  561 

.15  508 

.84492 

.00  439 

.84931 

52 

9 

.15  157 

.99  559 

.15  598 

.84  402 

.00  441 

.84  843 

51 

10 

9.15  245 

9.99  557 

9.15  688 

0.84  312 

0.00  443 

0.84  755 

50 

11 

.15  333 

.99  556 

.15  777 

.84  223 

.00  444 

.84  667 

49 

12 

.15421 

.99  554 

.15  867 

.84  133 

.00  446 

.84579 

48 

13 

.15508 

.99  552 

.15  956 

.84  044 

.00  448 

.84  492 

47 

14 

.15  596 

.99  550 

.16  046 

.83  954 

.00  450 

.84404 

46 

15 

9.15  683 

9.99  548 

9.16  135 

0.83  865 

0.00  452 

0.84  317 

45 

16 

.15  770 

.99  546 

.16  224 

.83  776 

.00  454 

.84230 

44 

17 

.15857 

.99  545 

.16312 

.83  688 

.00  455 

.84  143 

43 

18 

.15944 

.99  543 

.16401 

.83  599 

.00  457 

.84056 

42 

19 

.16  030 

.99  541 

.16  489 

.83511 

.00  459 

.83  970 

41 

20 

9.16  116 

9.99  539 

9.16  577 

0.83  423 

0.00  461 

0.83  884 

40 

21 

.16  203 

.99  537 

.16  665 

.83  335 

.00  463 

.83  797 

39 

22 

.16289 

.99  535 

.16  753 

.83  247 

.00  465 

.83  711 

38 

23 

.16  374 

.99  533 

.16  841 

.83  159 

.00  467 

.83  626 

37 

24 

.16460 

.99  532 

.16  928 

.83  072 

.00  468 

.83  540 

36 

25 

9.16  545 

9.99  530 

9.17016 

0.82  984 

0.00  470 

0.83  455 

35 

26 

.16631 

.99  528 

.17  103 

.82  897 

.00  472 

.83  369 

34 

27 

.16716 

.99  526 

.17  190 

.82  810 

.00  474 

.83  284 

33 

28 

.16801 

.99524 

.17277 

.82  723 

.00  476 

.83199 

32 

29 

.16  886 

.99  522 

.17363 

.82  637 

.00  478 

.83  114 

31 

30 

9.16  970 

9.99  520 

9.17450 

0.82  550 

0.00  480 

0.83  030 

30 

31 

.17  055 

.99  518 

.17  536 

.82  464 

.00  482 

.82  945 

29 

32 

.17  139 

.99  517 

17622 

.82  378 

.00  483 

.82  861 

28 

33 

.17  223 

.99  515 

.17  708 

.82  292 

.00  485 

.82  777 

27 

34 

.17  307 

.99  513 

.17  794 

.82  206 

.00  487 

.82  693 

26 

35 

9.17391 

9.99511 

9.17  880 

0.82  120 

0.00  489 

0.82  609 

25 

36 

.17474 

.99  509 

.17  965 

.82  035 

.00  491 

.82  526 

24 

37 

.17  558 

.99  507 

.18051 

.81  949 

.00  493 

.82  442 

23 

38 

.17  641 

.99  505 

.18  136 

.81  864 

.00  495 

.82  359 

22 

39 

.17  724 

.99  503 

.18221 

.81  779 

.00  497 

.82  276 

21 

40 

9.17  807 

9.99  501 

9.18306 

0.81  694 

0.00  499 

0.82  193 

20 

41 

.17  890 

.99  499 

.18391 

.81  609 

.00  501 

.82  110 

19 

42 

.17  973 

.99  497 

.18475 

.81  525 

.00  503 

.82  027 

18 

43 

.18  055 

.99  495 

.18  560 

.81  440 

.00  505 

.81  945 

17 

44 

.18  137 

.99  494 

.18  644 

.81  356 

.00  506 

.81  863 

16 

45 

9.18  220 

9.99  492 

9.18  728 

0.81  272 

0.00  508 

0.81  780 

15 

46' 

.18  302 

.99  490 

.18812 

.81  188 

.00  510 

.81  698 

14 

47 

.18  383 

.99  488 

.18896 

.81  104 

.00  512 

.81  617 

13 

48 

.18465 

.99  486 

.18979 

.81  021 

.00  514 

.81  535 

12 

49 

.18  547 

.99484 

.19  063 

.80  937 

.00  516 

.81  453 

11 

50 

9.18  628 

9.99  482 

9.19  146 

0.80  854 

0.00  518 

0.81  372 

10 

51 

.18  709 

.99  480 

.19  229 

.80  771 

.00  520 

.81  291 

9 

52 

.18  790 

.99  478 

.19312 

.80  688 

.00  522 

.81  210 

8 

53 

.18871 

.99  476 

.19  395 

.80  605 

.00  524 

.81  129 

7 

54 

.18  952 

.99  474 

.19478 

.80  522 

.00  526 

.81  048 

6 

55 

9.19  033 

9.99  472 

9.19  561 

0.80  439 

0.00  528 

0.80  967 

5 

56 

.19  113 

.99  470 

.19  643 

.80  357 

.00  530 

.80  887 

4 

57 

.19  193 

.99  468 

.19  725 

.80  275 

.00  532 

.80  807 

3 

58 

.19  273 

.99  466 

.19  807 

.80  193 

.00  534 

.80  727 

2 

59 

.19  353 

.99  464 

.19  889 

.80  111 

.00  536 

.80  647 

1 

60 

9.19433 

9.99  462 

9.19971 

0.80  029 

0.00  538 

0.80  567 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

98°  (278°) 


(261°)  81 c 


Table  4.    Trigonometric  Logarithms 


205 


9°  (189°) 


(350°)  170C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.19  433 

9.99  462 

9.19971 

0.80  029 

0.00  538 

0.80  567 

60 

1 

.19513 

.99  460 

.20  053 

.79  947 

.00  540 

.80  487 

59 

2 

.19  592 

.99  458 

.20  134 

.79  866 

.00542 

.80  408 

58 

3 

.19  672 

.99  456 

.20  216 

.79  784 

.00544 

.80  328 

57 

4 

.19  751 

.99  454 

.20  297 

.79  703 

.00546 

.80  249 

56 

5 

9.19  830 

9.99  452 

9.20  378 

0.79  622 

0.00  548 

0.80  170 

55 

6 

.19  909 

.99  450 

.20  459 

.79  541 

.00  550 

.80  091 

54 

7 

.19  988 

.99  448 

.20  540 

.79  460 

.00  552 

.80  012 

53 

8 

.20  067 

.99  446 

.20  621 

.79  379 

.00  554 

.79  933 

52 

9 

.20  145 

.99  444 

.20  701 

.79  299 

.00  556 

.79  855 

51 

10 

9.20  223 

9.99  442 

9.20  782 

0.79  218 

0.00  558 

0.79  777 

50 

11 

.20  302 

.99  440 

.20  862 

.79  138 

.00  560 

.79  698 

49 

12 

.20  380 

.99  438 

.20  942 

.79  058 

.00562 

.79  620 

48 

13 

.20  458 

.99  436 

.21  022 

.78  978 

.00  564 

.79  542 

47 

14 

.20  535 

.99  434 

.21  102 

.78  898 

.00  566 

.79  465 

46 

15 

9.20  613 

9.99  432 

9.21  182 

0.78  818 

0.00  568 

0.79  387 

45 

16 

.20  691 

.99  429 

.21  261 

.78  739 

.00  571 

.79  309 

44 

17 

.20  768 

.99  427 

.21  341 

.78  659 

.00573 

.79  232 

43 

18 

.20845 

.99  425 

.21  420 

.78  580 

.00  575 

.79  155 

42 

19 

.20  922 

.99  423 

.21  499 

.78  501 

.00  577 

.79  078 

41 

20 

9.20  999 

9.99  421 

9.21  578 

0.78  422 

0.00  579 

0.79  001 

40 

21 

.21  076 

.99419 

.21  657 

.78  343 

.00  581 

.78  924 

39 

22 

.21  153 

.99  417 

.21  736 

.78  264 

.00  583 

.78847 

38 

23 

.21  229 

.99  415 

.21  814 

.78  186 

.00  585 

.78  771 

37 

24 

.21  306 

.99  413 

.21  893 

.78  107 

.00  587 

.78  694 

36 

25 

9.21  382 

9.99411 

9.21  971 

0.78  029 

0.00  589 

0.78  618 

35 

26 

.21  458 

.99  409 

.22  049 

.77  951 

.00  591 

.78  542 

34 

27 

.21  534 

.99  407 

.22  127 

.77  873 

.00593 

.78  466 

33 

28 

.21  610 

.99  404 

.22  205 

.77  795 

.00  596 

.78  390 

32 

29 

.21  685 

.99  402 

.22  283 

.77  717 

.00  598 

.78  315 

31 

30 

9.21  761 

9.99  400 

9.22  361 

0.77  639 

0.00  600 

0.78  239 

30 

31 

.21  836 

.99  398 

.22  438 

.77  562 

.00  602 

.78  164 

29 

32 

.21  912 

.99  396 

.22  516 

.77  484 

.00  604 

.78  088 

28 

33 

.21  987 

.99  394 

.22  593 

.77  407 

.00  606 

.78  013 

27 

34 

.22  062 

.99  392 

.22  670 

.77  330 

.00  608 

.77  938 

26 

35 

9.22  137 

9.99  390 

9.22  747 

0.77  253 

0.00  610 

0.77  863 

25 

36 

.22211 

.99  388 

.22  824 

.77  176 

.00  612 

.77  789 

24 

37 

.22  286 

.99  385 

.22  901 

.77  099 

.00  615 

.77  714 

23 

38 

.22  361 

.99  383 

.22  977 

.77  023 

.00617 

.77  639 

22 

39 

.22  435 

.99  381 

.23  054 

.76  946 

.00619 

.77  565 

21 

40 

9.22  509 

9.99  379 

9.23  130 

0.76  870 

0.00  621 

0.77  491 

20 

41 

.22  583 

.99  377 

.23  206 

.76  794 

.00  623 

.77417 

19 

42 

.22  657 

.99  375 

.23  283 

.76  717 

.00  625 

.77  343 

18 

43 

.22  731 

.99  372 

.23  359 

.76  641 

.00  628 

.77  269 

17 

44 

.22  805 

.99  370 

.23  435 

.76  565 

.00  630 

.77  195 

16 

45 

9.22  878 

9.99  368 

9.23  510 

0.76  490 

0.00  632 

0.77  122 

15 

46 

.22  952 

.99  366 

.23  586 

.76  414 

.00  634 

.77  048 

14 

47 

.23  025 

.99364 

.23  661 

.76  339 

.00636 

.76  975 

13 

48 

.23  098 

.99  362 

.23  737 

.76  263 

.00  638 

.76  902 

12 

49 

.23  171 

.99  359 

.23  812 

.76  188 

.00641 

.76  829 

11 

50 

9.23  244 

9.99  357 

9.23  887 

0.76  113 

0.00  643 

0.76  756 

10 

51 

.23  317 

.99  355 

.23  962 

.76  038 

.00645 

.76  683 

9 

52 

.23  390 

.99  353 

.24  037 

.75  963 

.00647 

.76  610 

8 

53 

.23  462 

.99  351 

.24  112 

.75  888 

.00  649 

.76  538 

7 

54 

.23  535 

.99  348 

.24  186 

.75  814 

.00652 

.76  465 

6 

55 

9.23  607 

9.99  346 

9.24  261 

0.75  739 

0.00  654 

0.76  393 

5 

56 

.23  679 

.99  344 

.24  335 

.75  665 

.00  656 

.76  321 

4 

57 

.23  752 

.99  342 

.24  410 

.75  590 

.00  658 

.76  248 

3 

58 

.23  823 

.99  340 

.24484 

.75  516 

.00  660 

.76  177 

2 

59 

.23  895 

.99  337 

.24  558 

.75  442 

.00663 

.76  105 

1 

60 

9.23  967 

9.99  335 

9.24  632 

0.75  368 

0.00  665 

0.76  033 

0 

Cos 

Sin      Cot      Tan 

Csc      Sec 

99°  (279°) 


(260°)  80C 


206 


Table  4:.    Trigonometric  Logarithms 


10°  (190°) 


(349°)  169C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.23  967 

9.99  335 

9.24  632 

0.75  368 

0.00  665 

0.76  033 

60 

1 

.24  039 

.99  333 

.24  706 

.75  294 

.00  667 

.75  961 

59 

2 

.24  110 

.99  331 

.24  779 

.75  221 

.00  669 

.75  890 

58 

3 

.24  181 

.99  328 

.24  853 

.75  147 

.00  672 

.75  819 

57 

4 

.24  253 

.99  326 

.24  926 

.75  074 

.00  674 

.75  747 

56 

5 

9.24  324 

9.99  324 

9.25  000 

0.75  000 

0.00  676 

0.75  676 

55 

6 

.24  395 

.99  322 

.25  073 

.74  927 

.00  678 

.75  605 

54 

7 

.24  466 

.99  319 

.25  146 

.74  854 

.00  681 

.75  534 

53 

8 

.24  536 

.99317 

.25  219 

.74  781 

.00  683 

.75  464 

52 

9 

.24  607 

.99  315 

.25  292 

.74  708 

.00  685 

.75  393 

51 

10 

9.24  677 

9.99  313 

9.25  365 

0.74  635 

0.00  687 

0.75  323 

50 

11 

.24  748 

.99  310 

.25  437 

.74  563 

.00  690 

.75  252 

49 

12 

.24  818 

.99  308 

.25  510 

.74  490 

.00  692 

.75  182 

48 

13 

.24  888 

.99  306 

.25  582 

.74  418 

.00  694 

.75  112 

47 

14 

.24  958 

.99  304 

.25  655 

.74  345 

.00  696 

.75  042 

46 

15 

9.25  028 

9.99  301 

9.25  727 

0.74  273 

0.00  699 

0.74  972 

45 

16 

.25  098 

.99  299 

.25  799 

.74  201 

.00  701 

.74  902 

44 

17 

.25  168 

.99  297 

.25  871 

.74  129 

.00  703 

.74  832 

43 

18 

.25  237 

.99  294 

.25  943 

.74  057 

.00  706 

.74763 

42 

19 

.25  307 

.99  292 

.26  015 

.73  985 

.00  708 

.74  693 

41 

20 

9.25  376 

9.99  290 

9.26  086 

0.73  914 

0.00  710 

0.74  624 

40 

21 

.25  445 

.99  288 

.26  158 

.73  842 

.00  712 

.74  555 

39 

22 

.25  514 

.99  285 

.26  229 

.73  771 

.00  715 

.74  486 

38 

23 

.25  583 

.99  283 

.26  301 

.73  699 

.00  717 

.74  417 

37 

24 

.25  652 

.99  281 

.26  372 

.73  628 

.00  719 

.74  348 

36 

25 

9.25  721 

9.99  278 

9.26  443 

0.73  557 

0.00  722 

0.74  279 

35 

26 

.25  790 

.99  276 

.26  514 

.73  486 

.00  724 

.74  210 

34 

27 

.25  858 

.99  274 

.26  585 

.73415 

.00  726 

.74  142 

33 

28 

.25  927 

.99  271 

.26  655 

.73  345 

.00  729 

.74  073 

32 

29 

.25  995 

.99  269 

.26  726 

.73  274 

.00  731 

.74  005 

31 

30 

9.26  063 

9.99  267 

9.26  797 

0.73  203 

0.00  733 

0.73  937 

30 

31 

.26  131 

.99  264 

.26  867 

.73  133 

.00  736 

.73  869 

29 

32 

.26  199 

.99  262 

.26  937 

.73  063 

.00  738 

.73  801 

28 

33 

.26  267 

.99  260 

.27  008 

.72  992 

.00  740 

.73  733 

27 

34 

.26  335 

.99  257 

.27  078 

.72  922 

.00  743 

.73  665 

26 

35 

9.26  403 

9.99  255 

9.27  148 

0.72  852 

0.00  745 

0.73  597 

25 

36 

.26  470 

.99  252 

.27  218 

.72  782 

.00  748 

.73  530 

24 

37 

.26  538 

.99  250 

.27  288 

.72  712 

.00  750 

.73  462 

23 

38 

.26  605 

.99  248 

.27  357 

.72643 

.00  752 

.73  395 

22 

39 

.26  672 

.99  245 

.27  427 

.72  573 

.00  755 

.73  328 

21 

40 

9.26  739 

9.99  243 

9.27  496 

0.72  504 

0.00  757 

0.73  261 

20 

41 

.26  806 

.99  241 

.27  566 

.72  434 

.00  759 

.73  194 

19 

42 

.26  873 

.99  238 

.27  635 

.72  365 

.00  762 

.73  127 

18 

43 

.26  940 

.99  236 

.27  704 

.72  296 

.00  764 

.73  060 

17 

44 

.27  007 

.99  233 

.27  773 

.72  227 

.00  767 

.72  993 

16 

45 

9.27  073 

9.99  231 

9.27  842 

0.72  158 

0.00  769 

0.72  927 

15 

46 

.27  140 

.99  229 

.27911 

.72  089 

.00  771 

.72  860 

14 

47 

.27  206 

.99  226 

.27  980 

.72  020 

.00  774 

.72  794 

13 

48 

.27  273 

.99  224 

.28  049 

.71  951 

.00  776 

.72  727 

12 

49 

.27  339 

.99  221 

.28  117 

.71  883 

.00  779 

.72  661 

11 

50 

9.27  405 

9.99  219 

9.28  186 

0.71  814 

0.00  781 

0.72  595 

10 

51 

.27  471 

.99  217 

.28  254 

.71  746 

.00  783 

.72  529 

9 

52 

.27  537 

.99  214 

.28  323 

.71  677 

.00  786 

.72  463 

8 

53 

.27  602 

.99  212 

.28  391 

.71  609 

.00  788 

.72  398 

7 

54 

.27  668 

.99  209 

.28  459 

.71  541 

.00  791 

.72  332 

6 

55 

9.27  734 

9.99  207 

9.28  527 

0.71  473 

0.00  793 

0.72  266 

5 

56 

.27  799 

.99  204 

.28  595 

.71  405 

.00  796 

.72  201 

4 

57 

.27  864 

.99  202 

.28  662 

.71  338 

.00  798 

.72  136 

3 

58 

.27  930 

.99  200 

.28  730 

.71  270 

.00  800 

.72  070 

2 

59 

.27  995 

.99  197 

.28  798 

.71  202 

.00  803 

.72  005 

1 

60 

9.28  060 

9.99  195 

9.28  865 

0.71  135 

0.00  805 

0.71  940 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

100°  (280°) 


(259°)  79° 


Table  4.    Trigonometric  Logarithms 


20< 


11°  (191°) 


(348°)  168C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.28  060 

9.99  195 

9.28  865 

0.71  135 

0.00  805 

0.71  940 

60 

1 

.28  125 

.99  192 

.28  933 

.71  067 

.00  808 

.71  875 

59 

2 

.28  190 

.99  190 

.29  000 

.71  000 

.00  810 

.71  810 

58 

3 

.28  254 

.99  187 

.29  067 

.70  933 

.00  813 

.71  746 

57 

4 

.28  319 

.99  185 

.29  134 

.70  866 

.00  815 

.71  681 

56 

5 

9.28  384 

9.99  182 

9.29  201 

0.70  799 

0.00  818 

0.71  616 

55 

6 

.28  448 

.99  180 

.29  268 

.70  732 

.00  820 

.71  552 

54 

7 

.28  512 

.99  177 

.29  335 

.70  665 

.00  823 

.71  488 

53 

8 

.28  577 

.99  175 

.29  402 

.70  598 

.00  825 

.71  423 

52 

9 

.28  641 

.99  172 

.29  468 

.70  532 

.00  828 

.71  359 

51 

10 

9.28  705 

9.99  170 

9.29  535 

0.70  465 

0.00  830 

0.71  295 

50 

11 

.28  769 

.99  167 

.29  601 

.70  399 

.00  833 

.71  231 

49 

12 

.28  833 

.99  165 

.29  668 

.70  332 

.00835 

.71  167 

48 

13 

.28  896 

.99  162 

.29  734 

.70  266 

.00  838 

.71  104 

47 

14 

.28  960 

.99  160 

.29  800 

.70  200 

.00840 

.71  040 

46 

15 

9.29  024 

9.99  157 

9.29  866 

0.70  134 

0.00  843 

0.70  976 

45 

16 

.29  087 

.99  155 

.29  932 

.70  068 

.00845 

.70  913 

44 

17 

.29  150 

.99  152 

.29  998 

.70  002 

.00  848 

.70  850 

43 

18 

.29  214 

.99  150 

.30  064 

.69  936 

.00  850 

.70  786 

42 

19 

.29  277 

.99  147 

.30  130 

.69  870 

.00853 

.70  723 

41 

20 

9.29  340 

9.99  145 

9.30  195 

0.69  805 

0.00  855 

0.70  660 

40 

21 

.29  403 

.99  142 

.30  261 

.69  739 

.00858 

.70  597 

39 

22 

.29  466 

.99  140 

.30  326 

.69  674 

.00  860 

.70  534 

38 

23 

.29  529 

.99  137 

.30  391 

.69  609 

.00  863 

.70  471 

37 

24 

.29  591 

.99  135 

.30  457 

.69  543 

.00865 

.70  409 

36 

25 

9.29  654 

9.99  132 

9.30  522 

0.69  478 

0.00  868 

0.70  346 

35 

26 

.29  716 

.99  130 

.30  587 

.69  413 

.00  870 

.70  284 

34 

27 

.29  779 

.99  127 

.30  652 

.69  348 

.00  873 

.70  221 

33 

28 

.29  841 

.99  124 

.30  717 

.69  283 

.00  876 

.70  159 

32 

29 

.29  903 

.99  122 

.30  782 

.69  218 

.00  878 

.70  097 

31 

30 

9.29  966 

9.99  119 

9.30  846 

0.69  154 

0.00  881 

0.70  034 

30 

31 

.30  028 

.99  117 

.30911 

.69  089 

.00  883 

.69  972 

29 

32 

.30  090 

.99  114 

.30  975 

.69  025 

.00  886 

.69  910 

28 

33 

.30  151 

.99  112 

.31  040 

.68  960 

.00  888 

.69849 

27 

34 

.30  213 

.99  109 

.31  104 

.68  896 

.00  891 

.69  787 

26 

35 

9.30  275 

9.99  106 

9.31  168 

0.68  832 

0.00  894 

0.69  725 

25 

36 

.30  336 

.99  104 

.31  233 

.68  767 

.00  896 

.69  664 

24 

37 

.30  398 

.99  101 

.31  297 

.68  703 

.00  899 

.69  602 

23 

38 

.30  459 

.99  099 

.31  361 

.68  639 

.00  901 

.69  541 

22 

39 

.30  521 

.99  096 

.31  425 

.68  575 

.00904 

.69  479 

21 

40 

9.30  582 

9.99  093 

9.31  489 

0.68511 

0.00  907 

0.69  418 

20 

41 

.30  643 

.99  091 

.31  552 

.68  448 

.00  909 

.69  357 

19 

42 

.30  704 

.99  088 

.31  616 

.68384 

.00  912 

.69  296 

18 

43 

.30  765 

.99  086 

.31  679 

.68  321 

.00  914 

.69  235 

17 

44 

.30  826 

.99  083 

.31  743 

.68  257 

.00917 

.69  174 

16 

45 

9.30  887 

9.99  080 

9.31  806 

0.68  194 

0.00  920 

0.69  113 

15 

46 

.30  947 

.99  078 

.31  870 

.68  130 

.00  922 

.69  053 

14 

47 

.31  008 

.99  075 

.31  933 

.68  067 

.00  925 

.68  992 

13 

48 

.31  068 

.99  072 

.31  996 

.68  004 

.00  928 

.68  932 

12 

49 

.31  129 

.99  070 

.32  059 

.67  941 

.00930 

.68  871 

11 

50 

9.31  189 

9.99  067 

9.32  122 

0.67  878 

0.00  933 

0.68811 

10 

51 

.31  250 

.99064 

.32  185 

.67  815 

.00  936 

.68  750 

9 

52 

.31  310 

.99  062 

.32  248 

.67  752 

.00  938 

.68  690 

8 

53 

.31  370 

.99  059 

.32311 

.67  689 

.00  941 

.68  630 

7 

54 

.31  430 

.99  056 

.32  373 

.67  627 

.00  944 

.68  570 

6 

55 

9.31  490 

9.99  054 

9.32  436 

0.67  564 

0.00  946 

0.68  510 

5 

56 

.31  549 

.99  051 

.32  498 

.67  502 

.00949 

.68  451 

4 

57 

.31  609 

.99  048 

.32  561 

.67  439 

.00  952 

.68  391 

3 

58 

.31  669 

.99  046 

.32  623 

.67  377 

.00  954 

.68  331 

2 

59 

.31  728 

.99  043 

.32  685 

.67  315 

.00957 

.68  272 

1 

60 

9.31  788 

9.99  040 

9.32  747 

0.67  253 

0.00  960 

0.68  212 

0 

Cos 

Sin 

Cot 

'fan 

Csc 

Sec 

' 

101°  (281°) 


(258°)  78° 


208 


Table  4.    Trigonometric  Logarithms 


12°  (192°) 


(347°)   167 c 


' 

Sin 

Cos 

Tan 

Cot 

Sec    |   Csc 

0 

9.31  788 

9.99  040 

9.32  747 

0.67  253 

0.00  960 

0.68212 

60 

1 

.31  847 

.99  038 

.32  810 

.67  190 

.00  962 

.68  153 

59 

2 

.31  907 

.99  035 

.32  872 

.67  128 

.00  965 

.68  093 

58 

3 

.31  966 

.99  032 

.32  933 

.67  067 

.00  968 

.68  034 

57 

4 

.32  025 

.99  030 

.32  995 

.67  005 

.00  970 

.67  975 

56 

5 

9.32  084 

9.99  027 

9.33  057 

0.66  943 

0.00  973 

0.67  916 

55 

6 

.32  143 

.99  024 

.33  119 

.66  881 

.00  976 

.67  857 

54 

7 

.32  202 

.  .99  022 

.33  180 

.66  820 

.00  978 

.67  798 

53 

8 

.32  261 

.99  019 

.33  242 

.66  758 

.00  981 

.67  739 

52 

9 

.32  319 

.99  016 

.33  303 

.66  697 

.00  984 

.67  681 

51 

10 

9.32  378 

9.99  013 

9.33  365 

0.66  635 

0.00  987 

0.67  622 

50 

11 

.32  437 

.99011 

.33  426 

.66  574 

.00  989 

.67  563 

49 

12 

.32  495 

.99  008 

.33  487 

.66  513 

.00  992 

.67  505 

48 

13 

.32  553 

.99  005 

.33  548 

.66  452 

.00  995 

.67  447 

47 

14 

.32  612 

.99  002 

.33  609 

.66  391 

.00  998 

.67  388 

46 

15 

9.32  670 

9.99000 

9.33  670 

0.66  330 

0.01  000 

0.67  330 

45 

16 

.32  728 

.98  997 

.33  731 

.66  269 

.01  003 

.67  272 

44 

17 

.32  786 

.98  994 

.33  792 

.66  208 

.01  006 

.67  214 

43 

18 

.32  844 

.98  991 

.33  853 

.66  147 

.01  009 

.67  156 

42 

19 

.32  902 

.98  989 

.33  913 

.66  087 

.01  Oil 

.67  098 

41 

20 

9.32  960 

9.98  986 

9.33  974 

0.66  026 

0.01  014 

0.67  040 

40 

21 

.33  018 

.98  983 

.34  034 

.65  966 

.01  017 

.66  982 

39 

22 

.33  075 

.98  980 

.34  095 

.65  905 

.01  020 

.66  925 

38 

23 

.33  133 

.98  978 

.34  155 

.65  845 

.01  022 

.66  867 

37 

24 

.33  190 

.98  975 

.34  215 

.65  785 

.01  025 

.66  810 

36 

25 

9.33  248 

9.98  972 

9.34  276 

0.65  724 

0.01  028 

0.66  752 

35 

26 

.33  305 

.98  969 

.34  336 

.65  664 

.01  031 

.66  695 

34 

27 

.33  362 

.98  967 

.34  396 

.65  604 

.01  033 

.66  638 

33 

28 

.33  420 

.98  964 

.34  456 

.65  544 

.01  036 

.66  580 

32 

29 

.33  477 

.98  961 

.34  516 

.65  484 

.01  039 

.66  523 

31 

30 

9.33  534 

9.98  958 

9.34  576 

0.65  424 

0.01  042 

0.66  466 

30 

31 

.33  591 

.98  955 

.34  635 

.65  365 

.01  045 

.66  409 

29 

32 

.33  647 

.98  953 

.34  695 

.65  305 

.01  047 

.66  353 

28 

33 

.33  704 

.98  950 

.34  755 

.65  245 

.01  050 

.66  296 

27 

34 

.33  761 

.98  947 

.34  814 

.65  186 

.01  053 

.66  239 

26 

35 

9.33  818 

9.98  944 

9.34  874 

0.65  126 

0.01  056 

0.66  182 

25 

36 

.33  874 

.98  941 

.34  933 

.65  067 

.01  059 

.66  126 

24 

37 

.33  931 

.98  938 

.34  992 

.65  008 

.01  062 

.66  069 

23 

38 

.33  987 

.98  936 

.35  051 

.64  949 

.01  064 

.66  013 

22 

39 

.34  043 

.98  933 

.35  111 

.64  889 

.01  067 

.65  957 

21 

40 

9.34  100 

9.98  930 

9.35  170 

0.64  830 

0.01  070 

0.65  900 

20 

41 

.34  156 

.98  927 

.35  229 

.64  771 

.01  073 

.65  844 

19 

42 

.34  212 

.98  924 

.35  288 

.64  712 

.01  076 

.65  788 

18 

43 

.34  268 

.98  921 

.35  347 

.64  653 

.01  079 

.65  732 

17 

44 

.34  324 

,98  919 

.35  405 

.64  595 

.01  081 

.65  676 

16 

45 

9.34  380 

9.98  916 

9.35  464 

0.64  536 

0.01  084 

0.65  620 

15 

46 

.34  436 

.98  913 

.35  523 

.64  477 

.01  087 

.65  564 

14 

47 

.34  491 

.98  910 

.35  581 

.64  419 

.01  090 

.65  509 

13 

48 

.34  547 

.98  907 

.35  640 

.64  360 

.01  093 

.65  453 

12 

49 

.34  602 

.98  904 

.35  698 

.64  302 

.01  096 

.65  398 

11 

50 

9.34  658 

9.98  901 

9.35  757 

0.64  243 

0.01  099 

0.65  342 

10 

51 

.34  713 

.98  898 

.35  815 

.64  185 

.01  102 

.65  287 

9 

52 

.34  769 

.98  896 

.35  873 

.64  127 

.01  104 

.65  231 

8 

53 

.34  824 

.98  893 

.35  931 

.64  069 

.01  107 

.65  176 

7 

54 

.34  879 

.98  890 

.35  989 

.64011 

.01  110 

.65  121 

6 

55 

9.34  934 

9.98  887 

9.36  047 

0.63  953 

0.01  113 

0.65  066 

5 

56 

.34  989 

.98  884 

.36  105 

.63  895 

.01  116 

.65011 

4 

57 

.35  044 

.98  881 

.36  163 

.63  837 

.01  119 

.64  956 

3 

58 

.35  099 

.98  878 

.36  221 

.63  779 

.01  122 

.64  901 

2 

59 

.35  154 

.98  875 

.36  279 

.63  721 

.01  125 

.64  846 

1 

60 

9.35  209 

9.98  872 

9.36  336 

0.63  664 

0.01  128 

0.64  791 

0 

Cos 

Sin 

<Jot 

Tan 

Csc 

Sec 

> 

102°  (282°) 


(257°)  77° 


Table  4.    Trigonometric  Logarithms 


209 


13°  (193°) 


(346°)  166C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.35  209 

9.98  872 

9.36  336 

0.63  664 

0.01  128 

0.64  791 

60 

1 

.35  263 

.98  869 

.36  394 

.63  606 

.01  131 

.64  737 

59 

2 

.35  318 

.98  867 

.36  452 

.63  548 

.01  133 

.64  682 

58 

3 

.35  373 

.98864 

.36  509 

.63  491 

.01  136 

.64  627 

57 

4 

.35  427 

.98  861 

.36  566 

.63  434 

.01  139 

.64  573 

56 

5 

9.35  481 

9.98  858 

9.36  624 

0.63  376 

0.01  142 

0.64  519 

55 

6 

.35  536 

.98  855 

.36  681 

.63  319 

.01  145 

.64464 

54 

7 

.35  590 

.98  852 

.36  738 

.63  262 

.01  148 

.64  410 

53 

8 

.35  644 

.98849 

.36  795 

.63  205 

.01  151 

.64  356 

52 

9 

.35  698 

.98846 

.36  852 

.63  148 

.01  154 

.64302 

51 

10 

9.35  752 

9.98  843 

9.36  909 

0.63  091 

0.01  157 

0.64  248 

50 

11 

.35  806 

.98840 

.36  966 

.63  034 

.01  160 

.64  194 

49 

12 

.35  860 

.98  837 

.37  023 

.62  977 

.01  163 

.64  140 

48 

13 

.35  914 

.98  834 

.37  080 

.62  920 

.01  166 

.64  086 

47 

14 

.35  968 

.98  831 

.37  137 

.62  863 

.01  169 

.64  032 

46 

15 

9.36  022 

9.98  828 

9.37  193 

0.62  807 

0.01  172 

0.63  978 

45 

16 

.36  075 

.98  825 

.37  250 

.62  750 

.01  175 

.63  925 

44 

17 

.36  129 

.98  822 

.37  306 

.62  694 

.01  178 

.63  871 

43 

18 

.36  182 

.98  819 

.37  363 

.62  637 

.01  181 

.63  818 

42 

19 

.36  236 

.98  816 

.37  419 

.62  581 

.01  184 

.63  764 

41 

20 

9.36  289 

9.98  813 

9.37  476 

0.62  524 

0.01  187 

0.63  711 

40 

21 

.36  342 

.98  810 

.37  532 

.62  468 

.01  190 

.63  658 

39 

22 

.36  395 

.98  807 

.37  588 

.62  412 

.01  193 

.63  605 

38 

23 

.36  449 

.98  804 

.37  644 

.62  356 

.01  196 

.63  551 

37 

24 

.36  502 

.98  801 

.37  700 

.62  300 

.01  199 

.63  498 

36 

25 

9.36  555 

9.98  798 

9.37  756 

0.62  244 

0.01  202 

0.63  445 

35 

26 

.36  608 

.98  795 

.37  812 

.62  188 

.01  205 

.63  392 

34 

27 

.36  660 

.98  792 

.37  868 

.62  132 

.01  208 

.63  340 

33 

28 

.36  713 

.98  789 

.37  924 

.62  076 

.01  211 

.63  287 

32 

29 

.36  766 

.98  786 

.37  980 

.62  020 

.01  214 

.63  234 

31 

30 

9.36  819 

9.98  783 

9.38  035 

0.61  965 

0.01  217 

0.63  181 

30 

31 

.36  871 

.98  780 

.38  091 

.61  909 

.01  220 

.63  129 

29 

32 

.36  924 

.98  777 

.38  147 

.61  853 

.01  223 

.63  076 

28 

33 

.36  976 

.98  774 

.38  202 

.61  798 

.01  226 

.63  024 

27 

34 

.37  028 

.98  771 

.38  257 

.61  743 

.01  229 

.62  972 

26 

35 

9.37  081 

9.98  768 

9.38  313 

0.61  687 

0.01  232 

0.62  919 

25 

36 

.37  133 

.98  765 

.38  368 

.61  632 

.01  235 

.62  867 

24 

37 

.37  185 

.98  762 

.38  423 

.61  577 

.01  238 

.62  815 

23 

38 

.37  237 

.98  759 

.38  479 

.61  521 

.01  241 

.62  763 

22 

39 

.37  289 

.98  756 

.38  534 

.61  466 

.01  244 

.62711 

21 

40 

9.37  341 

9.98  753 

9.38  589 

0.61  411 

0.01  247 

0.62  659 

20 

41 

.37  393 

.98  750 

.38644 

.61  356 

.01  250 

.62  607 

19 

42 

.37445 

.98  746 

.38  699 

.61  301 

.01  254 

.62  555 

18 

43 

.37  497 

.98  743 

.38  754 

.61  246 

.01  257 

.62  503 

17 

44 

.37  549 

.98  740 

.38  808 

.61  192 

.01  260 

.62  451 

16 

45 

9.37  600 

9.98  737 

9.38  863 

0.61  137 

0.01  263 

0.62  400 

15 

46 

.37  652 

.98  734 

.38  918 

.61  082 

.01  266 

.62  348 

14 

47 

.37  703 

.98  731 

.38  972 

.61  028 

.01  269 

.62  297 

13 

48 

.37  755 

.98  728 

.39  027 

.60  973 

.01  272 

.62  245 

12 

49 

.37  806 

.98  725 

.39  082 

.60  918 

.01  275 

.62  194 

11 

50 

9.37  858 

9.98  722 

9.39  136 

0.60  864 

0.01  278 

0.62  142 

10 

51 

.37  909 

.98  719 

.39  190 

.60  810 

.01  281 

.62  091 

9 

52 

.37  960 

.98  715 

.39  245 

.60  755 

.01285 

.62040 

8 

53 

.38011 

.98  712 

.39  299 

.60  701 

.01  288 

.61  989 

7 

54 

.38  062 

.98  709 

.39  353 

.60  647 

.01  291 

.61  938 

6 

55 

9.38  113 

9.98  706 

9.39  407 

0.60  593 

0.01  294 

0.61  887 

5 

56 

.38  164 

.98  703 

.39  461 

.60  539 

.01  297 

.61  836 

4 

57 

.38  215 

.98700 

.39  515 

.60  485 

.01  300 

.61785 

3 

58 

.38266 

.98  697 

.39  569 

.60  431 

.01  303 

.61  734 

2 

59 

.38  317 

.98  694 

.39  623 

.60  377 

.01  306 

.61  683 

1 

60 

9.38  368 

9.98  690 

9.39  677 

0.60  323 

0.01  310 

0.61  632 

0 

Cos      Sin 

Cot 

Tan 

Csc 

Sec 

' 

103°  (283°) 


(256°)  76° 


210 


Table  4.    Trigonometric  Logarithms 


14°  (194°) 


(345°)  165C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.38  368 

9.98  690 

9.39  677 

0.60  323 

0.01  310 

0.61  632 

60 

1 

.38418 

.98  687 

.39  731 

.60  269 

.01  313 

.61  582 

59 

2 

.38  469 

.98  684 

.39  785 

.60  215 

.0,1  316 

.61  531 

58 

3 

.38  519 

.98  681 

.39  838 

.60  162 

.01  319 

.61  481 

57 

4 

.38  570 

.98  678 

.39  892 

.60  108 

.01  322 

.61  430 

56 

5 

9.38  620 

9.98  675 

9.39  945 

0.60  055 

0.01  325 

0.61  380 

55 

6 

.38  670 

.98  671 

.39  999 

.60  001 

.01  329 

.61  330 

54 

7 

.38  721 

.98  668 

.40  052 

.59  948 

.01  332 

.61  279 

53 

8 

.38  771 

.98  665 

.40  106 

.59  894 

.01  335 

.61  229 

52 

9 

.38  821 

.98  662 

.40  159 

.59  841 

.01  338 

.61  179 

51 

10 

9.38  871 

9.98  659 

9.40  212 

0.59  788 

0.01  341 

0.61  129 

50 

11 

.38  921 

.98  656 

.40  266 

.59  734 

.01  344 

.61  079 

49 

12 

.38  971 

.98  652 

.40  319 

.59  681 

.01  348 

.61  029 

48 

13 

.39  021 

.98  649 

.40  372 

.59  628 

.01  351 

.60  979 

47 

14 

.39  071 

.98  646 

.40  425 

.59  575 

.01  354 

.60  929 

46 

15 

9.39  121 

9.98  643 

9.40  478 

0.59  522 

0.01  357 

0.60  879 

45 

16 

.39  170 

.98  640 

.40  531 

.59  469 

.01  360 

.60  830 

44 

17 

.39  220 

.98  636 

.40  584 

.59  416 

.01  364 

.60  780 

43 

18 

.39  270 

.98  633 

.40  636 

.59364 

.01  367 

.60  730 

42 

19 

.39  319 

.98  630 

.40  689 

.59  311 

.01  370 

.60  681 

41 

20 

9.39  369 

9.98  627 

9.40  742 

0.59  258 

0.01  373 

0.60  631 

40 

21 

.39  418 

.98  623 

.40  795 

.59  205 

.01  377 

.60  582 

39 

22 

.39  467 

.98  620 

.40  847 

.59  153 

.01  380 

.60  533 

38 

23 

.39  517 

.98  617 

.40  900 

.59  100 

.01  383 

.60  483 

37 

24 

.39  566 

.98  614 

.40  952 

.59  048 

.01  386 

.60  434 

36 

25 

9.39  615 

9.98  610 

9.41  005 

0.58995 

0.01  390 

0.60  385 

35 

26 

.39  664 

.98  607 

.41  057 

.58  943 

.01  393 

.60  336 

34 

27 

.39  713 

.98  604 

.41  109 

.58  891 

.01  396 

.60  287 

33 

28 

.39  762 

.98  601 

.41  161 

.58  839 

.01  399 

.60  238 

32 

29 

.39811 

.98  597 

.41  214 

.58  786 

.01  403 

.60  189 

31 

30 

9.39  860 

9.98  594 

9.41  266 

0.58  734 

0.01  406 

0.60  140 

30 

31 

.39  909 

.98  591 

.41  318 

.58  682 

.01  409 

.60  091 

29 

32 

.39  958 

.98  588 

.41  370 

.58  630 

.01  412 

.60  042 

28 

33 

.40  006 

.98  584 

.41  422 

.58  578 

.01  416 

.59  994 

27 

34 

.40  055 

.98  581 

.41  474 

.58  526 

.01  419 

.59  945 

26 

35 

9.40  103 

9.98  578 

9.41  526 

0.58  474 

0.01  422 

0.59  897 

25 

36 

.40  152 

.98  574 

.41  578 

.58  422 

.01  426 

.59  848 

24 

37 

.40  200 

.98  571 

.41  629 

.58  371 

.01  429 

.59  800 

23 

38 

.40  249 

.98  568 

.41  681 

.58  319 

.01  432 

.59  751 

22 

39 

.40  297 

.98  565 

.41  733 

.58  267 

.01  435 

.59  703 

21 

40 

9.40  346 

9.98  561 

9.41  784 

0.58  216 

0.01  439 

0.59  654 

20 

41 

.40  394 

.98  558 

.41  836 

.58  164 

.01  442 

.59  606 

19 

42 

.40  442 

.98  555 

.41  887 

.58  113 

.01  445 

.59  558 

18 

43 

.40  490 

.98  551 

.41  939 

.58  061 

.01  449 

.59  510 

17 

44 

.40  538 

.98  548 

.41  990 

.58  010 

.01  452 

.59  462 

16 

45 

9.40  586 

9.98  545 

9.42  041 

0.57  959 

0.01  455 

0.59  414 

15 

46 

.40  634 

.98  541 

.42  093 

.57  907 

.01  459 

.59  366 

14 

47 

.40  682 

.98  538 

.42  144 

.57  856 

.01  462 

.59  318 

13 

48 

.40  730 

.98  535 

.42  195 

.57  805 

.01465 

.59  270 

12 

49 

.40  778 

.98  531 

.42  246 

.57  754 

.01  469 

.59  222 

11 

50 

9.40  825 

9.98  528 

9.42  297 

0.57  703 

0.01  472 

0.59  175 

10 

51 

.40  873 

.98  525 

.42  348 

.57  652 

.01  475 

.59  127 

9 

52 

.40  921 

.98  521 

.42  399 

.57  601 

.01  479 

.59  079 

8 

53 

.40  968 

.98518 

.42  450 

.57  550 

.01  482 

.59  032 

7 

54 

.41  016 

.98515 

.42  501 

.57  499 

.01  485 

.58984 

6 

55 

9.41  063 

9.98511 

9.42  552 

0.57  448 

0.01  489 

0.58  937 

5 

56 

.41  111 

.98  508 

.42  603 

.57  397 

.01  492 

.58  889 

4 

57 

.41  158 

.98  505 

.42  653 

.57  347 

.01  495 

.58  842 

3 

58 

.41  205 

.98  501 

.42  704 

.57  296 

.01  499 

.58  795 

2 

59 

.41  252 

.98  498 

.42  755 

.57  245 

.01  502 

.58  748 

1 

60 

9.41  300 

9.98  494 

9.42  805 

0.57  195 

0.01  506 

0.58  700 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

104°  (284°) 


(255°)  75° 


Table  4.    Trigonometric  Logarithms 


211 


15°  (195°) 


(344°)  164° 


' 

Sin 

Cos 

Tan      Cot 

Sec 

Csc 

0 

9.41  300 

9.98  494 

9.42  805 

0.57  195 

0.01  506 

0.58  700 

60 

1 

.41  347 

.98  491 

.42  856 

.57  144 

.01  509 

.58  653 

59 

2 

.41  394 

.98  488 

.42  906 

.57  094 

.01  512 

.58  606 

58 

3 

.41  441 

.98484 

.42  957 

.57  043 

.01  516 

.58  559 

57 

4 

.41  488 

.98  481 

.43  007 

.56  993 

.01  519 

.58  512 

56 

5 

9.41  535 

9.98  477 

9.43  057 

0.56  943 

0.01  523 

0.58  465 

55 

6 

.41  582 

.98  474 

.43  108 

.56  892 

.01  526 

.58418 

54 

7 

.41  628 

.98  471 

.43  158 

.56842 

.01  529 

.58  372 

53 

8 

.41  675 

.98  467 

.43  208 

.56  792 

.01  533 

.58  325 

52 

9 

.41  722 

.98464 

.43  258 

.56  742 

.01  536 

.58  278 

51 

10 

9.41  768 

9.98  460 

9.43  308 

0.56  692 

0.01  540 

0.58  232 

50 

11 

.41  815 

.98  457 

.43  358 

.56  642 

.01  543 

.58  185 

49 

12 

.41  861 

.98  453 

.43  408 

.56  592 

.01  547 

.58  139 

48 

13 

.41  908 

.98  450 

.43  458 

.56  542 

.01  550 

.58  092 

47 

14 

.41  954 

.98447 

.43  508 

.56  492 

.01  553 

.58  046 

46 

15 

9.42  001 

9.98  443 

9.43  558 

0.56  442 

0.01  557 

0.57  999 

45 

16 

.42047 

.98440 

.43  607 

.56  393 

.01  560 

.57  953 

44 

17 

.42  093 

.98  436 

.43  657 

.56  343 

.01  564 

.57  907 

43 

18 

.42  140 

.98  433 

.43  707 

.56  293 

.01  567 

.57  860 

42 

19 

.42  186 

.98  429 

.43  756 

.56  244 

.01  571 

.57  814 

41 

20 

9.42  232 

9.98  426 

9.43  806 

0.56  194 

0.01  574 

0.57  768 

40 

21 

.42  278 

.98  422 

.43  855 

.56  145 

.01  578 

.57  722 

39 

22 

.42  324 

.98419 

.43  905 

.56  095 

.01  581 

.57  676 

38 

23 

.42  370 

.98  415 

.43  954 

.56046 

.01  585 

.57  630 

37 

24 

.42  416 

.98412 

.44  004 

.55  996 

.01  588 

.57  584 

36 

25 

9.42  461 

9.98  409 

9.44  053 

0.55  947 

0.01  591 

0.57  539 

35 

26 

.42  507 

.98  405 

.44  102 

.55  898 

.01  595 

.57  493 

34 

27 

.42  553 

.98  402 

.44  151 

.55849 

.01  598 

.57  447 

33 

28 

.42  599 

.98  398 

.44201 

.55  799 

.01  602 

.57  401 

32 

29 

.42  644 

.98  395 

.44250 

.55  750 

.01  605 

.57  356 

31 

30 

9.42  690 

9.98  391 

9.44  299 

0.55  701 

0.01  609 

0.57  310 

30 

31 

.42  735 

.98  388 

.44348 

.55  652 

.01  612 

.57  265 

29 

32 

.42  781 

.98384 

.44397 

.55  603 

.01  616 

.57  219 

28 

33 

.42  826 

.98  381 

.44446 

.55  554 

.01  619 

.57  174 

27 

34 

.42872 

.98  377 

.44495 

.55  505 

.01  623 

.57  128 

26 

35 

9.42  917 

9.98  373 

9.44  544 

0.55  456 

0.01  627 

0.57  083 

25 

36 

.42  962 

.98  370 

.44  592 

.55  408 

.01  630 

.57  038 

24 

37 

.43  008 

.98  366 

.44641 

.55  359 

.01  634 

.56  992 

23 

38 

.43  053 

.98  363 

.44  690 

.55  310 

.01  637 

.56  947 

22 

39 

.43  098 

.98  359 

.44738 

.55  262 

.01  641 

.56  902 

21 

40 

9.43  143 

9.98  356 

9.44  787 

0.55  213 

0.01  644 

0.56  857 

20 

41 

.43  188 

.98  352 

.44836 

.55  164 

.01  648 

.56  812 

19 

42 

.43  233 

.98  349 

.44884 

.55  116 

.01  651 

.56  767 

18 

43 

.43  278 

.98  345 

.44933 

.55  067 

.01  655 

.56  722 

17 

44 

.43  323 

.98  342 

.44981 

.55  019 

.01  658 

.56  677 

16 

45 

9.43  367 

9.98  338 

9.45  029 

0.54  971 

0.01  662 

0.56  633 

15 

46 

.43  412 

.98  334 

.45  078 

.54  922 

.01  666 

.56  588 

14 

47 

.43  457 

.98  331 

.45  126 

.54  874 

.01  669 

.56  543 

13 

48 

.43  502 

.98  327 

.45  174 

.54826 

.01  673 

.56  498 

12 

49 

.43546 

.98  324 

.45  222 

.54778 

.01  676 

.56  454 

11 

50 

9.43  591 

9.98  320 

9.45  271 

0.54  729 

0.01  680 

0.56  409 

10 

51 

.43  635 

.98  317 

.45  319 

.54  681 

.01  683 

.56  365 

9 

52 

.43  680 

.98  313 

.45  367 

.54  633 

.01  687 

.56  320 

8 

53 

.43  724 

.98  309 

.45  415 

.54  585 

.01  691 

.56  276 

7 

54 

.43  769 

.98  306 

.45  463 

.54  537 

.01  694 

.56  231 

6 

55 

9.43  813 

9.98  302 

9.45511 

0.54  489 

0.01  698 

0.56  187 

5 

56 

.43  857 

.98  299 

.45  559 

.54441 

.01  701 

.56  143 

4 

57 

.43  901 

.98  295 

.45  606 

.54  394 

.01  705 

.56  099 

3 

58 

.43  946 

.98  291 

.45  654 

.54  346 

.01  709 

.56  054 

2 

59 

.43  990 

.98  288 

.45  702 

.54298 

.01  712 

.56  010 

1 

60 

9.44  034 

9.98  284 

9.45  750 

0.54  250 

0.01  716 

0.55  966 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

105°  (285°) 


(254°)  74° 


212 


Table  4.    Trigonometric  Logarithms 


16°  (196°) 


(343°)  163° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.44  034 

9.98  284 

9.45  750 

0.54  250 

0.01  716 

0.55  966 

60 

1 

.44  078 

.98  281 

.45  797 

.54  203 

.01  719 

.55  922 

59 

2 

.44  122 

.98  277 

.45  845 

.54  155 

.01  723 

.55  878 

58 

3 

.44  166 

.98  273 

.45  892 

.54  108 

.01  727 

.55  834 

57 

4 

.44  210 

.98  270 

.45  940 

.54  060 

.01  730 

.55  790 

56 

5 

9.44  253 

9.98  266 

9.45  987 

0.54  013 

0.01  734 

0.55  747 

55 

6 

.44  297 

.98  262 

.46  035 

.53  965 

.01  738 

.55  703 

54 

7 

.44  341 

.98  259 

.46  082 

.53  918 

.01  741 

.55  659 

53 

8 

.44  385 

.98  255 

.46  130 

.53  870 

.01  745 

.55  615 

52 

9 

.44  428 

.98  251 

.46  177 

.53  823 

.01  749 

.55  572 

51 

10 

9.44  472 

9.98  248 

9.46  224 

0.53  776 

0.01  752 

0.55  528 

50 

11 

.44  516 

.98  244 

.46  271 

.53  729 

.01  756 

.55484 

49 

12 

.44  559 

.98  240 

.46  319 

.53  681 

.01  760 

.55  441 

48 

13 

.44  602 

.98  237 

.46  366 

.53  634 

.01  763 

.55  398 

47 

14 

.44  646 

.98  233 

.46  413 

.53  587 

.01  767 

.55  354 

46 

15 

9.44  689 

9.98  229 

9.46  460 

0.53  540 

0.01  771 

0.55311 

45 

16 

.44  733 

.98  226 

.46  507 

.53  493 

.01  774 

.55  267 

44 

17 

.44  776 

.98  222 

.46  554 

.53  446 

.01  778 

.55  224 

43 

18 

.44  819 

.98218 

.46  601 

.53  399 

.01  782 

.55  181 

42 

19 

.44  862 

.98  215 

.46  648 

.53  352 

.01  785 

.55  138 

41 

20 

9.44  905 

9.98211 

9.46  694 

0.53  306 

0.01  789 

0.55  095 

40 

21 

.44  948 

.98  207 

.46  741 

.53  259 

.01  793 

.55  052 

39 

22 

.44  992 

.98  204 

.46  788 

.53  212 

.01  796 

.55  008 

38 

23 

.45  035 

.98  200 

.46  835 

.53  165 

.01  800 

.54  965 

37 

24 

.45  077 

.98  196 

.46  881 

.53  119 

.01  804 

.54  923 

36 

25 

9.45  120 

9.98  192 

9.46  928 

0.53  072 

0.01  808 

0.54  880 

35 

26 

.45  163 

.98  189 

.46  975 

.53  025 

.01  811 

.54  837 

34 

27 

.45  206 

.98  185 

.47  021 

.52  979 

.01  815 

.54  794 

33 

28 

.45  249 

.98  181 

.47  068 

.52  932 

.01  819 

.54  751 

32 

29 

.45  292 

.98  177 

.47  114 

.52  886 

.01  823 

.54  708 

31 

30 

9.45  334 

9.98  174 

9.47  160 

0.52  840 

0.01  826 

0.54  666 

30 

31 

.45  377 

.98  170 

.47  207 

.52  793 

.01  830 

.54  623 

29 

32 

.45  419 

.98  166 

.47  253 

.52  747 

.01  834 

.54  581 

28 

33 

.45  462 

.98  162 

.47  299 

.52  701 

.01  838 

.54  538 

27 

34 

.45  504 

.98  159 

.47  346 

.52  654 

.01  841 

.54  496 

26 

35 

9.45  547 

9.98  155 

9.47  392 

0.52  608 

0.01  845 

0.54  453 

25 

36 

.45  589 

.98  151 

.47  438 

.52  562 

.01  849 

.54411 

24 

37 

.45  632 

.98  147 

.47  484 

.52  516 

.01  853 

.54  368 

23 

38 

.45  674 

.98  144 

.47  530 

.52  470 

.01  856 

.54  326 

22 

39 

.45  716 

.98  140 

.47  576 

.52  424 

.01  860 

.54  284 

21 

40 

9.45  758 

9.98  136 

9.47  622 

0.52  378 

0.01  864 

0.54  242 

20 

41 

.45  801 

.98  132 

.47  668 

.52  332 

.01  868 

.54  199 

19 

42 

.45  843 

.98  129 

.47  714 

.52  286 

.01  871 

.54  157 

18 

43 

.45  885 

.98  125 

.47  760 

.52  240 

.01  875 

.54  115 

17 

44 

.45  927 

.98  121 

.47  806 

.52  194 

.01  879 

.54  073 

16 

45 

9.45  969 

9.98  117 

9.47  852 

0.52  148 

0.01  883 

0.54  031 

15 

46 

.46011 

.98  113 

.47  897 

.52  103 

.01  887 

.53  989 

14 

47 

.46  053 

.98  110 

.47  943 

.52  057 

.01  890 

.53  947 

13 

48 

.46  095 

.98  106 

.47  989 

.52011 

.01  894 

.53  905 

12 

49 

.46  136 

.98  102 

.48  035 

.51  965 

.01  898 

.53  864 

11 

50 

9.46  178 

9.98  098 

9.48  080 

0.51  920 

0.01  902 

0.53  822 

10 

51 

.46  220 

.98  094 

.48  126 

.51  874 

.01  906 

.53  780 

9 

52 

.46  262 

.98  090 

.48  171 

.51  829 

.01  910 

.53  738 

8 

53 

.46  303 

.98  087 

.48  217 

.51  783 

.01  913 

.53  697 

7 

54 

.46  345 

.98  083 

.48  262 

.51  738 

.01  917 

.53  655 

6 

55 

9.46  386 

9.98  079 

9.48  307 

0.51  693 

0.01  921 

0.53  614 

5 

56 

.46  428 

.98  075 

.48  353 

.51  647 

.01  925 

.53  572 

4 

57 

.46  469 

.98  071 

.48  398 

.51  602 

.01  929 

.53  531 

3 

58 

.46511 

.98  067 

.48  443 

.51  557 

.01  933 

.53  489 

2 

59 

.46  552 

.98  063 

.48  489 

.51  511 

.01  937 

.53  448 

1 

60 

9.46  594 

9.98  060 

9.48  534 

0.51  466 

0.01  940 

0.53  406 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

106°  (286°) 


(253°)  73° 


Table  4.    Trigonometric  Logarithms 


213 


17°  (197°) 


(342°)  162C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.46  594 

9.98  060 

9.48  534 

0.51  466 

0.01  940 

0.53  406 

60 

1 

.46  635 

.98  056 

.48  579 

.51  421 

.01  944 

.53  365 

59 

2 

.46  676 

.98  052 

.48  624 

.51  376 

.01  948 

.53  324 

58 

3 

.46  717 

.98  048 

.48  669 

.51  331 

.01  952 

.53  283 

57 

4 

.46  758 

.98044 

.48  714 

.51  286 

.01  956 

.53  242 

56 

5 

9.46  800 

9.98  040 

9.48  759 

0.51  241 

0.01  960 

0.53  200 

55 

6 

.46841 

.98  036 

.48  804 

.51  196 

.01  964 

.53  159 

54 

7 

.46  882 

.98  032 

.48849 

.51  151 

.01  968 

.53  118 

53 

8 

.46  923 

.98  029 

.48  894 

.51  106 

.01  971 

.53  077 

52 

9 

.46964 

.98  025 

.48  939 

.51  061 

.01  975 

.53  036 

51 

10 

9.47  005 

9.98  021 

9.48  984 

0.51  016 

0.01  979 

0.52  995 

50 

11 

.47045 

.98  017 

.49  029 

.50  971 

.01  983 

.52  955 

49 

12 

.47  086 

.98  013 

.49  073 

.50  927 

.01  987 

.52  914 

48 

13 

.47  127 

'  .98009 

.49  118 

.50  882 

.01  991 

.52  873 

47 

14 

.47  168 

.98  005 

.49  163 

.50  837 

.01  995 

.52  832 

46 

15 

9.47  209 

9.98  001 

9.49  207 

0.50  793 

0.01  999 

0.52  791 

45 

16 

.47  249 

.97  997 

.49  252 

.50  748 

.02  003 

.52  751 

44 

17 

.47  290 

.97  993 

.49  296 

.50  704 

.02  007 

.52  710 

43 

18 

.47  330 

.97  989 

.49  341 

.50  659 

.02011 

.52  670 

42 

19 

.47  371 

.97  986 

.49  385 

.50  615 

.02  014 

.52  629 

41 

20 

9.47411 

9.97  982 

9.49  430 

0.50  570 

0.02  018 

0.52  589 

40 

21 

.47  452 

.97  978 

.49  474 

.50  526 

.02  022 

.52  548 

39 

22 

.47  492 

.97  974 

.49  519 

.50  481 

.02  026 

.52  508 

38 

23 

.47  533 

.97  970 

.49  563 

.50  437 

.02  030 

.52  467 

37 

24 

.47  573 

.97  966 

.49  607 

.50  393 

.02  034 

.52  427 

36 

25 

9.47  613 

9.97  962 

9.49  652 

0.50  348 

0.02  038 

0.52  387 

35 

26 

.47  654 

.97  958 

.49  696 

.50  304 

.02042 

.52  346 

34 

27 

.47  694 

.97  954 

.49  740 

.50  260 

.02  046 

.52  306 

33 

28 

.47  734 

.97  950 

.49784 

.50  216 

.02  050 

.52  266 

32 

29 

.47  774 

.97  946 

.49  828 

.50  172 

.02  054 

.52  226 

31 

30 

9.47  814 

9.97  942 

9.49  872 

0.50  128 

0.02  058 

0.52  186 

30 

31 

.47  854 

.97  938 

.49  916 

.50084 

.02  062 

.52  146 

29 

32 

.47  894 

.97  934 

.49  960 

.50  040 

.02  066 

.52  106 

28 

33 

.47  934 

.97  930 

.50  004 

.49  996 

.02  070 

.52  066 

27 

34 

.47  974 

.97  926 

.50  048 

.49  952 

.02  074 

.52  026 

26 

35 

9.48  014 

9.97  922 

9.50  092 

0.49  908 

0.02  078 

0.51  986 

25 

36 

.48  054 

.97  918 

.50  136 

.49  864 

.02  082 

.51  946 

24 

37 

.48  094 

.97  914 

.50  180 

.49  820 

.02  086 

.51  906 

23 

38 

.48  133 

.97  910 

.50  223 

.49  777 

.02  090 

.51  867 

22 

39 

.48  173 

.97  906 

.50  267 

.49  733 

.02  094 

.51  827 

21 

40 

9.48  213 

9.97  902 

9.50311 

0.49  689 

0.02  098 

0.51  787 

20 

41 

.48  252 

.97  898 

.50  355 

.49  645 

.02  102 

.51  748 

19 

42 

.48  292 

.97  894 

.50  398 

.49  602 

.02  106 

.51  708 

18 

43 

.48  332 

.97  890 

.50  442 

.49  558 

.02  110 

.51  668 

17 

44 

.48  371 

.97  886 

.50  485 

.49  515 

.02  114 

.51  629 

16 

45 

9.48411 

9.97  882 

9.50  529 

0.49  471 

0.02  118 

0.51  589 

15 

46 

.48  450 

.97  878 

.50  572 

.49  428 

.02  122 

.51  550 

14 

47 

.48  490 

.97  874 

.50  616 

.49384 

.02  126 

.51  510 

13 

48 

.48  529 

.97  870 

.50  659 

.49  341 

.02  130 

.51  471 

12 

49 

.48  568 

.97  866 

.50  703 

.49  297 

.02  134 

.51  432 

11 

50 

9.48  607 

9.97  861 

9.50  746 

0.49  254 

0.02  139 

0.51  393 

10 

51 

.48  647 

.97  857 

.50  789 

.49  211 

.02  143 

.51  353 

9 

52 

.48  686 

.97853 

.50833 

.49  167 

.02  147 

.51  314 

8 

53 

.48  725 

.97849 

.50  876 

.49  124 

.02  151 

.51  275 

7 

54 

.48  764 

.97845 

.50  919 

.49  081 

.02  155 

.51  236 

6 

55 

9.48  803 

9.97  841 

9.50  962 

0.49  038 

0.02  159 

0.51  197 

5 

56 

.48842 

.97  837 

.51  005 

.48  995 

.02  163 

.51  158 

4 

57 

.48  881 

.97  833 

.51048 

.48  952 

.02  167 

.51  119 

3 

58 

.48  920 

.97  829 

.51  092 

.48  908 

.02  171 

.51  080 

2 

59 

.48  959 

.97  825 

.51  135 

.48  865 

.02  175 

.51  041 

1 

60 

9.48  998 

9.97  821 

9.51  178 

0.48  822 

0.02  179 

0.51  002 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

107°  (287°) 


(252°)  72° 


214 


Table  4.    Trigonometric  Logarithms 


18°  (198°) 


(341°)  161° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.48  998 

9.97  821 

9.51  178 

0.48  822 

0.02  179 

0.51  002 

60 

1 

.49  037 

.97  817 

.51  221 

.48  779 

.02  183 

.50  963 

59 

2 

.49  076 

.97  812 

.51  264 

.48  736 

.02  188 

.50  924 

58 

3 

.49  115 

.97  808 

.51  306 

.48  694 

.02  192 

.50  885 

57 

4 

.49  153 

.97  804 

.51  349 

.48  651 

.02  196 

.50  847 

56 

5 

9.49  192 

9.97  800 

9.51  392 

0.48  608 

0.02  200 

0.50  808 

55 

6 

.49  231 

.97  796 

.51  435 

.48  565 

.02  204 

.50  769 

54 

7 

.49  269 

.97  792 

.51  478 

.48  522 

.02  208 

.50  731 

53 

8 

.49  308 

.97  788 

.51  520 

.48  480 

.02  212 

.50  692 

52 

9 

.49  347 

.97784 

.51  563 

.48  437 

.02  216 

.50  653 

51 

10 

9.49  385 

9.97  779 

9.51  606 

0.48  394 

0.02  221 

0.50  615 

50 

11 

.49  424 

.97  775 

.51  648 

.48  352 

.02  225 

.50  576 

49 

12 

.49  462 

.97  771 

.51  691 

.48  309 

.02  229 

.50  538 

48 

13 

.49  500 

.97  767 

.51  734 

.48  266 

.02  233 

.50  500 

47 

14 

.49  539 

.97  763 

.51  776 

.48  224 

.02  237 

.50  461 

46 

15 

9.49  577 

9.97  759 

9.51  819 

0.48  181 

0.02  241 

0.50  423 

45 

16 

.49  615 

.97  754 

.51  861 

.48  139 

.02  246 

.50  385 

44 

17 

.49  654 

.97  750 

.51  903 

.48  097 

.02  250 

.50  346 

43 

18 

.49  692 

.97  746 

.51  946 

.48  054 

.02  254 

.50  308 

42 

19 

,49  730 

.97  742 

.51  988 

.48  012 

.02  258 

.50  270 

41 

20 

9.49  768 

9.97  738 

9.52  031 

0.47  969 

0.02  262 

0.50  232 

40 

21 

.49  806 

.97  734 

.52  073 

.47  927 

.02  266 

.50  194 

39 

22 

.49844 

.97  729 

.52  115 

.47  885 

.02  271 

.50  156 

38 

23 

.49  882 

.97  725 

.52  157 

.47843 

.02  275 

.50  118 

37 

24 

.49  920 

.97  721 

.52  200 

.47  800 

.02  279 

.50  080 

36 

25 

9.49  958 

9.97  717 

9.52  242 

0.47  758 

0.02  283 

0.50  042 

35 

26 

.49  996 

.97  713 

.52284 

.47  716 

.02  287 

.50  004 

34 

27 

.50  034 

.97  708 

.52  326 

.47  674 

.02  292 

.49  966 

33 

28 

.50  072 

.97  704 

.52  368 

.47  632 

.02  296 

.49  928 

32 

29 

.50  110 

.97  700 

.52  410 

.47  590 

.02  300 

.49  890 

31 

30 

9.50  148 

9.97  696 

9.52  452 

0.47  548 

0.02  304 

0.49  852 

30 

31 

.50  185 

.97  691 

.52  494 

.47  506 

.02  309 

.49  815 

29 

32 

.50  223 

.97  687 

.52  536 

.47  464 

.02  313 

.49  777 

28 

33 

.50  261 

.97  683 

.52  578 

.47  422 

.02  317 

.49  739 

27 

34 

.50  298 

.97  679 

.52  620 

.47  380 

.02  321 

.49  702 

26 

35 

9.50  336 

9.97  674 

9.52  661 

0.47  339 

0.02  326 

0.49  664 

25 

36 

.50  374 

.97  670 

.52  703 

.47  297 

.02  330 

.49  626 

24 

37 

.50411 

.97  666 

.52  745 

.47255 

.02  334 

.49  5,89 

23 

38 

.50  449 

.97  662 

.52  787 

.47  213 

.02  338 

.49  551 

22 

39 

.50  486 

.97  657 

.52  829 

.47  171 

.02  343 

.49  514 

21 

40 

9.50  523 

9.97  653 

9.52  870 

0.47  130 

0.02  347 

0.49  477 

20 

41 

.50  561 

.97  649 

.52912 

.47  088 

.02  351 

.49  439 

19 

42 

.50  598 

.97645 

.52  953 

.47  047 

.02  355 

.49  402 

18 

43 

.50  635 

.97  640 

.52  995 

.47  005 

.02  360 

.49  365 

17 

44 

.50  673 

.97  636 

.53  037 

.46  963 

.02  364 

.49  327 

16 

45 

9.50  710 

9.97  632 

9.53  078 

0.46  922 

0.02  368 

0.49  290 

15 

46 

.50  747 

.97  628 

.53  120 

.46  880 

.02  372 

.49  253 

14 

47 

.50  784 

.97  623 

.53  161 

.46  839 

.02  377 

.49  216 

13 

48 

.50  821 

.97  619 

.53  202 

.46  798 

.02  381 

.49  179 

12 

49 

.50  858 

.97  615 

.53  244 

.46  756 

.02  385 

.49  142 

11 

50 

9.50  896 

9.97  610 

9.53  285 

0.46  715 

0.02  390 

0.49  104 

10 

51 

.50  933 

.97  606 

.53  327 

.46  673 

.02  394 

.49  067 

9 

52 

.50  970 

.97  602 

.53  368 

.46  632 

.02  398 

.49  030 

8 

53 

.51  007 

.97  597 

.53  409 

.46  591 

.02  403 

.48  993 

7 

54 

.51  043 

.97  593 

.53  450 

.46  550 

.02  407 

.48  957 

6 

55 

9.51  080 

9.97  589 

9.53  492 

0.46  508 

0.02411 

0.48  920 

5 

56 

.51  117 

.97  584 

.53  533 

.46  467 

.02  416 

.48  883 

4 

57 

.51  154 

.97  580 

.53  574 

.46  426 

.02  420 

.48  846 

3 

58 

.51  191 

.97  576 

.53  615 

.46  385 

.02  424 

.48  809 

2 

59 

.51  227 

.97  571 

.53  656 

.46  344 

.02  429 

.48  773 

1 

60 

9.51  264 

9.97  567 

9.53  697 

0.46  303 

0.02  433 

0.48  736 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

108°  (288°) 


(251°)  71C 


Table  4.    Trigonometric  Logarithms 


215 


19°  (199°) 


(340°)  160° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.51  264 

9.97  567 

9.53  697 

0.46  303 

0.02  433 

0.48  736 

60 

1 

.51  301 

.97  563 

.53  738 

.46  262 

.02  437 

.48  699 

59 

2 

.51  338 

.97  558 

.53  779 

.46  221 

.02442 

.48  662 

58 

3 

.51  374 

.97  554 

.53  820 

.46  180 

.02  446 

.48  626 

57 

4 

.51411 

.97  550 

.53  861 

.46  139 

.02  450 

.48  589 

56 

5 

9.51  447 

9.97  545 

9.53  902 

0.46  098 

0.02  455 

0.48  553 

55 

6 

.51  484 

.97  541 

.53  943 

.46  057 

.02  459 

.48  516 

54 

7 

.51  520 

.97  536 

.53  984 

.46  016 

.02  464 

.48  480 

53 

8 

.51  557 

.97  532 

.54  025 

.45  975 

.02  468 

.48  443 

52 

9 

.51  593 

.97  528 

.54  065 

.45  935 

.02  472 

.48  407 

51 

10 

9.51  629 

9.97  523 

9.54  106 

0.45  894 

0.02  477 

0.48  371 

50 

11 

.51  666 

.97  519 

.54  147 

.45  853 

.02  481 

.48  334 

49 

12 

.51  702 

.97  515 

.54187 

.45  813 

.02  485 

.48  298 

48 

13 

.51  738 

.97  510 

.54  228 

.45  772 

.02  490 

.48  262 

47 

14 

.51  774 

.97  506 

.54  269 

.45  731 

.02  494 

.48  226 

46 

15 

9.51  811 

9.97  501 

9.54  309 

0.45  691 

0.02  499 

0.48  189 

45 

16 

.51  847 

.97  497 

.54350 

.45  650 

.02  503 

.48  153 

44 

17 

.51  883 

.97  492 

.54  390 

.45  610 

.02  508 

.48  117 

43 

18 

.51  919 

.97  488 

.54  431 

.45  569 

.02  512 

.48  081 

42 

19 

.51  955 

.97484 

.54  471 

.45  529 

.02  516 

.48  045 

41 

20 

9.51  991 

9.97  479 

9.54  512 

0.45  488 

0.02  521 

0.48  009 

40 

21 

.52  027 

.97  475 

.54  552 

.45  448 

.02  525 

.47  973 

39 

22 

.52  063 

.97  470 

.54  593 

.45  407 

.02  530 

.47  937 

38 

23 

.52  099 

.97  466 

.54  633 

.45  367 

.02  534 

.47  901 

37 

24 

.52  135 

.97  461 

.54  673 

.45  327 

.02  539 

.47  865 

36 

25 

9.52  171 

9.97  457 

9.54  714 

0.45  286 

0.02  543 

0.47  829 

35 

26 

.52  207 

.97  453 

.54  754 

.45  246 

.02  547 

.47  793 

34 

27 

.52  242 

.97448 

.54  794 

.45  206 

.02  552 

.47  758 

33 

28 

.52  278 

.97  444 

.54  835 

.45  165 

.02  556 

.47  722 

32 

29 

.52  314 

.97  439 

.54  875 

.45  125 

.02  561 

.47  686 

31 

30 

9.52  350 

9.97  435 

9.54  915 

0.45  085 

0.02  565 

0.47  650 

30 

31 

.52  385 

.97  430 

.54  955 

.45  045 

.02  570 

.47  615 

29 

32 

.52  421 

.97  426 

.54  995 

.45  005 

.02  574 

.47  579 

28 

33 

.52  456 

.97  421 

.55  035 

.44965 

.02  579 

.47  544 

27 

34 

.52  492 

.97  417 

.55  075 

.44  925 

.02  583 

.47  508 

26 

35 

9.52  527 

9.97  412 

9.55  115 

0.44  885 

0.02  588 

0.47  473 

25 

36 

.52  563 

.97  408 

.55  155 

.44845 

.02  592 

.47  437 

24 

37 

.52  598 

.97  403 

.55  195 

.44  805 

.02  597 

.47  402 

23 

38 

.52  634 

.97  399 

.55  235 

.44765 

.02  601 

.47  366 

22 

39 

.52  669 

.97  394 

.55  275 

.44725 

.02  606 

.47  331 

21 

40 

9.52  705 

9.97  390 

9.55  315 

0.44  685 

0.02  610 

0.47  295 

20 

41 

.52  740 

.97  385 

.55  355 

.44645 

.02  615 

.47  260 

19 

42 

.52  775 

.97  381 

.55  395 

.44605 

.02  619 

.47  225 

18 

43 

.52811 

.97  376 

.55  434 

.44566 

.02  624 

.47  189 

•  17 

44 

.52846 

.97  372 

.55  474 

.44526 

.02  628 

.47  154 

16 

45 

9.52  881 

9.97  367 

9.55  514 

0.44  486 

0.02  633 

0.47  119 

15 

46 

.52  916 

.97  363 

.55554 

.44446 

.02  637 

.47084 

14 

47 

.52  951 

.97  358 

.55  593 

.44407 

.02642 

.47  049 

13 

48 

.52  986 

.97  353 

.55  633 

.44367 

.02647 

.47  014 

12 

49 

.53  021 

.97  349 

.55  673 

.44327 

.02  651 

.46  979 

11 

50 

9.53  056 

9.97  344 

9.55  712 

0.44  288 

0.02  656 

0.46  944 

10 

51 

.53  092 

.97  340 

.55  752 

.44248 

.02  660 

.46  908 

9 

52 

.53  126 

.97  335 

.55  791 

.44209 

.02  665 

.46  874 

8 

53 

.53  161 

.97  331 

.55  831 

.44169 

.02  669 

.46  839 

7 

54 

.53  196 

.97  326 

.55  870 

.44130 

.02  674 

.46804 

6 

55 

9.53  231 

9.97  322 

9.55  910 

0.44  090 

0.02  678 

0.46  769 

5 

56 

.53  266 

.97  317 

.55  949 

.44051 

.02683 

.46  734 

4 

57 

.53  301 

.97  312 

.55  989 

.44011 

.02  688 

.46  699 

3 

58 

.53  336 

.97  308 

.56  028 

.43  972 

.02  692 

.46664 

2 

59 

.53  370 

.97  303 

.56  067 

.43  933 

.02  697 

.46  630 

1 

60 

9.53  405 

9.97  299 

9.56  107 

9.43  893 

0.02  701 

0.46  595 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

109°  (289°) 


(250°)  70° 


216 


Table  4.    Trigonometric  Logarithms 


20°  (200°) 


(339°)  159° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.53  405 

9.97  299 

9.56  107 

0.43  893 

0.02  701 

0.46  595 

60 

1 

.53  440 

.97  294 

.56  146 

.43  854 

.02  706 

.46  560 

59 

2 

.53  475 

.97  289 

.56  185 

.43  815 

.02711 

.46  525 

58 

3 

.53  509 

.97  285 

.56  224 

.43  776 

.02  715 

.46  491 

57 

4 

.53  544 

.97  280 

.56  264 

.43  736 

.02  720 

.46  456 

56 

5 

9.53  578 

9.97  276 

9.56  303 

0.43  697 

0.02  724 

0.46  422 

55 

6 

.53  613 

.97  271 

.56  342 

.43  658 

.02  729 

.46  387 

54 

7 

.53  647 

.97  266 

.56  381 

.43  619 

.02  734 

.46  353 

53 

8 

.53  682 

.97  262 

.56  420 

.43  580 

.02  738 

.46  318 

52 

9 

.53  716 

.97  257 

.56  459 

.43  541 

.02  743 

.46  284 

51 

10 

9.53  751 

9.97  252 

9.56  498 

0.43  502 

0.02  748 

0.46  249 

50 

11 

.53  785 

.97  248 

.56  537 

.43  463 

.02  752 

.46  215 

49 

12 

.53  819 

.97  243 

.56  576 

.43  424 

.02  757 

.46  181 

48 

13 

.53  854 

.97  238 

.56  615 

.43  385 

.02  762 

.46  146 

47 

14 

.53  888 

.97  234 

.56  654 

.43  346 

.02  766 

.46  112 

46 

15 

9.53  922 

9.97  229 

9.56  693 

0.43  307 

0.02  771 

0.46  078 

45 

16 

.53  957 

.97  224 

.56  732 

.43  268 

.02  776 

.46  043 

44 

17 

.53  991 

.97  220 

.56  771 

.43  229 

.02  780 

.46  009 

43 

18 

.54  025 

.97  215 

.56  810 

.43  190 

.02  785 

.45  975 

42 

19 

.54  059 

.97  210 

.56  849 

.43  151 

.02  790 

.45  941 

41 

20 

9.54  093 

9.97  206 

9.56  887 

0.43  113 

0.02  794 

0.45  907 

40 

21 

.54  127 

.97  201 

.56  926 

.43  074 

.02  799 

.45  873 

39 

22 

.54  161 

.97  196 

.56  965 

.43  035 

.02  804 

.45  839 

38 

23 

.54  195 

.97  192 

.57  004 

.42  996 

.02  808 

.45  805 

37 

24 

.54  229 

.97  187 

.57  042 

.42  958 

.02  813 

.45  771 

36 

25 

9.54  263 

9.97  182 

9.57  081 

0.42  919 

0.02  818 

0.45  737 

35 

26 

.54  297 

.97  178 

.57  120 

.42  880 

.02  822 

.45  703 

34 

27 

.54  331 

.97  173 

.57  158 

.42842 

.02  827 

.45  669 

33 

28 

.54  365 

.97  168 

.57  197 

.42  803 

.02  832 

.45  635 

32 

29 

.54  399 

.97  163 

.57  235 

.42  765 

.02  837 

.45  601 

31 

30 

9.54  433 

9.97  159 

9.57  274 

0.42  726 

0.02  841 

0.45  567 

30 

31 

.54  466 

.97  154 

.57  312 

.42  688 

.02846 

.45  534 

29 

32 

.54  500 

.97  149 

.57  351 

.42  649 

.02  851 

.45  500 

28 

33 

.54  534 

.97  145 

.57  389 

.42  611 

.02  855 

.45  466 

27 

34 

.54  567 

.97  140 

.57  428 

.42  572 

.02  860 

.45  433 

26 

35 

9.54  601 

9.97  135 

9.57  466 

0.42  534 

0.02  865 

0.45  399 

25 

36 

.54  635 

.97  130 

.57  504 

.42  496 

.02  870 

.45  365 

24 

37 

.54  668 

.97  126 

.57  543 

.42  457 

.02  874 

.45  332 

23 

38 

.54  702 

.97  121 

.57  581 

.42419 

.02  879 

.45  298 

22 

39 

.54  735 

.97  116 

.57  619 

.42  381 

.02  884 

.45  265 

21 

40 

9.54  769 

9.97  111 

9.57  658 

0.42  342 

0.02  889 

0.45  231 

20 

41 

.54  802 

.97  107 

.57  696 

.42  304 

.02  893 

.45  198 

19 

42 

.54  836 

.97  102 

.57  734 

.42  266 

.02  898 

.45  164 

18 

43 

.54  869 

.97  097 

'  .57  772 

.42  228 

.02  903 

.45  131 

17 

44 

.54  903 

.97  092 

.57  810 

.42  190 

.02  908 

.45  097 

16 

45 

9.54  936 

9.97  087 

9.57  849 

0.42  151 

0.02  913 

0.45  064 

15 

46 

.54  969 

.97  083 

.57  887 

.42  113 

.02  917 

.45  031 

14 

47 

.55  003 

.97  078 

.57  925 

.42  075 

.02  922 

.44  997 

13 

48 

.55  036 

.97  073 

.57  963 

.42  037 

.02  927 

.44  964 

12 

49 

.55  069 

.97  068 

.58  001 

.41  999 

.02  932 

.44  931 

11 

50 

9.55  102 

9.97  063 

9.58  039 

0.41  961 

0.02  937 

0.44  898 

10 

51 

.55  136 

.97  059 

.58  077 

.41  923 

.02  941 

.44864 

9 

52 

.55  169 

.97  054 

.58  115 

.41  885 

.02  946 

.44  831 

8 

53 

.55  202 

.97  049 

.58  153 

.41  847 

.02  951 

.44  798 

7 

54 

.55  235 

.97  044 

.58  191 

.41  809 

.02  956 

.44  765 

6 

55 

9.55  268 

9.97  039 

9.58  229 

0.41  771 

0.02  961 

0.44  732 

5 

56 

.55  301 

.97  035 

.58  267 

.41  733 

.02  965 

.44  699 

4 

57 

.55  334 

.97  030 

.58  304 

.41  696 

.02  970 

.44  666 

3 

58 

.55  367 

.97  025 

.58  342 

.41  658 

.02  975 

.44633 

2 

59 

.55  400 

.97  020 

.58  380 

.41  620 

.02  980 

.44  600 

1 

60 

9.55  433 

9.97  015 

9.58418 

0.41  582 

0.02  985 

0.44  567 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

110°  (290°) 


(249°)  69C 


Table  4.    Trigonometric  Logarithms 


217 


21°  (201°) 


(338°)  158C 


/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.55  433 

9.97  015 

9.58418 

0.41  582 

0.02  985 

0.44  567 

60 

1 

.55  466 

.97  010 

.58  455 

.41  545 

.02  990 

.44  534 

59 

2 

.55  499 

.97  005 

.58  493 

.41  507 

.02  995 

.44  501 

58 

3 

.55  532 

.97  001 

.58  531 

.41  469 

.02  999 

.44  468 

57 

4 

.55  564 

.96  996 

.58  569 

.41  431 

.03  004 

.44  436 

56 

5 

9.55  597 

9.96  991 

9.58  606 

0.41  394 

0.03  009 

0.44  403 

55 

6 

.55  630 

.96  986 

.58644 

.41  356 

.03  014 

.44  370 

54 

7 

.55  663 

.96  981 

.58  681 

.41  319 

.03  019 

.44  337 

53 

8 

.55  695 

.96  976 

.58  719 

.41  281 

.03  024 

.44  305 

52 

9 

.55  728 

.96  971 

.58  757 

.41  243 

.03  029 

.44  272 

51 

10 

9.55  761 

9.96  966 

9.58  794 

0.41  206 

0.03  034 

0.44  239 

50 

11 

.55  793 

.96  962 

.58  832 

.41  168 

.03  038 

.44  207 

49 

12 

.55  826 

.96  957 

.58  869 

.41  131 

.03  043 

.44174 

48 

•  13 

.55  858 

.96  952 

.58  907 

.41  093 

.03048 

.44142 

47 

14 

.55  891 

.96  947 

.58944 

.41  056 

.03  053 

.44109 

46 

15 

9.55  923 

9.96  942 

9.58  981 

0.41  019 

0.03  058 

0.44  077 

45 

16 

.55  956 

.96  937 

.59  019 

.40  981 

.03  063 

.  .44044 

44 

17 

.55  988 

.96  932 

.59  056 

.40  944 

.03  068 

.44  012 

43 

18 

.56  021 

.96  927 

.59  094 

.40  906 

.03  073 

.43  979 

42 

19 

.56  053 

.96  922 

.59  131 

.40  869 

.03  078 

.43  947 

41 

20 

9.56  085 

9.96  917 

9.59  168 

0.40  832 

0.03  083 

0.43  915 

40 

21 

.56  118 

.96  912 

.59  205 

.40  795 

.03  088 

.43  882 

39 

22 

.56  150 

.96  907 

.59  243 

.40  757 

.03  093 

.43  850 

38 

23 

.56  182 

.96  903 

.59  280 

.40  720 

.03  097 

.43  818 

37 

24 

.56215 

.96  898 

.59  317 

.40  683 

.03  102 

.43  785 

36 

25 

9.56  247 

9.96  893 

9.59  354 

0.40  646 

0.03  107 

0.43  753 

35 

26 

.56  279 

.96  888 

.59  391 

.40  609 

.03  112 

.43  721 

34 

27 

.56311 

.96  883 

.59  429 

.40  571 

.03  117 

.43  689 

33 

28 

.56  343 

.96  878 

.59  466 

.40  534 

.03  122 

.43  657 

32 

29 

.56  375 

.96  873 

.59  503 

.40  497 

.03  127 

.43  625 

31 

30 

9.56  408 

9.96  868 

9.59  540 

0.40  460 

0.03  132 

0.43  592 

30 

31 

.56440 

.96  863 

.59  577 

.40  423 

.03  137 

.43  560 

29 

32 

.56  472 

.96  858 

.59  614 

.40  386 

.03  142 

.43  528 

28 

33 

.56  504 

.96853 

.59  651 

.40  349 

.03  147 

.43  496 

27 

34 

.56  536 

.96848 

.59  688 

.40  312 

.03  152 

.43  464 

26 

35 

9.56  568 

9.96  843 

9.59  725 

0.40  275 

0.03  157 

0.43  432 

25 

36 

.56  599 

.96  838 

.59  762 

.40  238 

.03  162 

.43  401 

24 

37 

.56  631 

.96  833 

.59  799 

.40  201 

.03  167 

.43  369 

23 

38 

.56  663 

.96  828 

.59  835 

.40  165 

.03  172 

.43  337 

22 

39 

.56  695 

.96  823 

.59  872 

.40  128 

.03  177 

.43  305 

21 

40 

9.56  727 

9.96  818 

9.59  909 

0.40  091 

0.03  182 

0.43  273 

20 

41 

.56  759 

.96  813 

.59  946 

.40  054 

.03  187 

.43  241 

19 

42 

.56  790 

.96  808 

.59  983 

.40017 

.03  192 

.43  210 

18 

43 

.56  822 

.96  803 

.60  019 

.39  981 

.03  197 

.43  178 

17 

44 

.56854 

.96  798 

.60  056 

.39  944 

.03  202 

.43  146 

16 

45 

9.56  886 

9.96  793 

9.60  093 

0.39  907 

0.03  207 

0.43  114 

15 

46 

.56  917 

.96  788 

.60  130 

.39  870 

.03  212 

.43  083 

14 

47 

.56  949 

.96  783 

.60  166 

.39  834 

.03  217 

.43  051 

13 

48 

.56  980 

.96  778 

.60  203 

.39  797 

.03  222 

.43  020 

12 

49 

.57  012 

.96  772 

.60  240 

.39  760 

.03  228 

.42  988 

11 

50 

9.57  044 

9.96  767 

9.60  276 

0.39  724 

0.03  233 

0.42  956 

10 

51 

.57  075 

.96  762 

.60  313 

.39  687 

.03  238 

.42  925 

9 

52 

.57  107 

.96  757 

.60  349 

.39  651 

.03  243 

.42  893 

8 

53 

.57  138 

.96  752 

.60  386 

.39  614 

.03  248 

.42  862 

7 

54 

.57  169 

.96  747 

.60  422 

.39  578 

.03  253 

.42  831 

6 

55 

9.57  201 

9.96  742 

9.60  459 

0.39  541 

0.03  258 

0.42  799 

5 

56 

.57  232 

.96  737 

.60  495 

.39  505 

.03263 

.42  768 

4 

57 

.57  264 

.96732 

.60  532 

.39  468 

.03  268 

.42  736 

3 

58 

.57  295 

.96  727 

.60  568 

.39  432 

.03  273 

.42  705 

2 

59 

.57  326 

.96  722 

.60  605 

.39  395 

.03  278 

.42  674 

1 

60 

9.57  358 

9.96  717 

9.60  641 

0.39  359 

0.03  283 

0.42  642 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

111°  (291°) 


(248°)  68° 


218 


Table  4.    Trigonometric  Logarithms 


22°  (202°) 


(337°)  157° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.57  358 

9.96  717 

9.60  641 

0.39  359 

0.03  283 

0.42  642 

60 

1 

.57  389 

.96711 

.60  677 

.39  323 

.03  289 

.42  611 

59 

2 

.57  420 

.96  706 

.60  714 

.39  286 

.03  294 

.42  580 

58 

3 

.57  451 

.96  701 

.60  750 

.39  250 

.03  299 

.42  549 

57 

4 

.57  482 

.96  696 

.60  786 

.39  214 

.03  304 

.42  518 

56 

5 

9.57  514 

9.96  691 

9.60  823 

0.39  177 

0.03  309 

0.42  486 

55 

6 

.57  545 

.96  686 

.60  859 

.39  141 

.03  314 

.42  455 

54 

7 

.57  576 

.96  681 

.60  895 

.39  105 

.03  319 

.42  424 

53 

8 

.57  607 

.96  676 

.60  931 

.39  069 

.03  324 

.42  393 

52 

9 

.57  638 

.96  670 

.60  967 

.39  033 

.03  330 

.42  362 

51 

10 

9.57  669 

9.96  665 

9.61  004 

9.38  996 

0.03  335 

0.42  331 

50 

11 

.57  700 

.96  660 

.61  040 

.38  960 

.03  340 

.42  300 

49 

12 

.57  731 

.96  655 

.61  076 

.38  924 

.03  345 

.42  269 

48 

13 

.57  762 

.96  650 

.61  112 

.38  888 

.03  350 

.42  238 

47 

•  14 

.57  793 

.96  645 

.61  148 

.38  852 

.03  355 

.42  207 

46 

15 

9.57  824 

9.96  640 

9.61  184 

0.38  816 

0.03  360 

0.42  176 

45 

16 

.57  855 

.96  634 

.61  220 

.38  780 

.03  366 

.42  145 

44 

17 

.57  885 

.96  629 

.61  256 

.38  744 

.03  371 

.42  115 

43 

18 

.57  916 

.96  624 

.61  292 

.38  708 

.03  376 

.42084 

42 

19 

.57  947 

.96  619 

.61  328 

.38  672 

.03  381 

.42  053 

41 

20 

9.57  978 

9.96  614 

9.61  364 

0.38  636 

0.03  386 

0.42  022 

40 

21 

.58  008 

.96  608 

.61  400 

.38  600 

.03  392 

.41  992 

39 

22 

.58  039 

.96  603 

.61  436 

.38  564 

.03  397 

.41  961 

38 

23 

.58  070 

.96  598 

.61  472 

.38  528 

.03  402 

.41  930 

37 

24 

.58  101 

.96  593 

.61  508 

.38  492 

.03  407 

.41  899 

36 

25 

9.58  131 

9.96  588 

9.61  544 

0.38  456 

0.03  412 

0.41  869 

35 

26 

.58  162 

.96  582 

.61  579 

.38  421 

.03  418 

.41  838 

34 

27 

.58  192 

.96  577 

.61  615 

.38  385 

.03  423 

.41  808 

33 

28 

.58  223 

.96  572 

.61  651 

.38  349 

.03  428 

.41  777 

32 

29 

.58  253 

.96  567 

.61  687 

.38313 

.03  433 

.41  747 

31 

30 

9.58  284 

9.96  562 

9.61  722 

0.38  278 

0.03  438 

0.41  716 

30 

31 

.58  314 

.96  556 

.61  758 

.38  242 

.03  444 

.41  686 

29 

32 

.58  345 

.96  551 

.61  794 

.38  206 

.03  449 

.41  655 

28 

33 

.58  375 

.96  546 

.61  830 

.38  170 

.03  454 

.41  625 

27 

34 

.58  406 

.96  541 

.61  865 

.38  135 

.03  459 

.41  594 

26 

35 

9.58  436 

9.96  535 

9.61  901 

0.38  099 

0.03  465 

0.41  564 

25 

36 

.58  467 

.96  530 

.61  936 

.38  064 

.03  470 

.41  533 

24 

37 

.58  497 

.96  525 

.61  972 

.38  028 

.03  475 

.41  503 

23 

38 

.58  527 

.96  520 

.62  008 

.37  992 

.03  480 

.41  473 

22 

39 

.58  557 

.96  514 

.62  043 

.37  957 

.03  486 

.41  443 

21 

40 

9.58  588 

9.96  509 

9.62  079 

0.37  921 

0.03  491 

0.41  412 

20 

41 

.58  618 

.96  504 

.62  114 

.37  886 

.03  496 

.41  382 

19 

42 

.58648 

.96  498 

.62  150 

.37  850 

.03  502 

.41  352 

18 

43 

.58  678 

.96  493 

.62  185 

.37  815 

.03  507 

.41  322 

17 

44 

.58  709 

.96  488 

.62  221 

.37  779 

.03  512 

.41  291 

16 

45 

9.58  739 

9.96  483 

9.62  256 

0.37  744 

0.03  517 

0.41  261 

15 

46 

.58  769 

.96  477 

.62  292 

.37  708 

.03  523 

.41  231 

14 

47 

.58  799 

.96  472 

.62  327 

.37  673 

.03  528 

.41  201 

13 

48 

.58  829 

.96  467 

.62  362 

.37  638 

.03  533 

.41  171 

12 

49 

.58  859 

.96  461 

.62  398 

.37  602 

.03  539 

.41  141 

11 

50 

9.58  889 

9.96  456 

9.62  433 

0.37  567 

0.03  544 

0.41  111 

10 

51 

.58  919 

.96  451 

.62  468 

.37  532 

.03  549 

.41  081 

9 

52 

.58  949 

.96  445 

.62  504 

.37  496 

.03  555 

.41  051 

8 

53 

.58  979 

.96  440 

.62  539 

.37  461 

.03  560 

.41  021 

7 

54 

.59  009 

.96  435 

.62  574 

.37  426 

.03  565 

.40  991 

6 

55 

9.59  039 

9.96  429 

9.62  609 

0.37  391 

0.03  571 

0.40  961 

5 

56 

.59  069 

.96  424 

.62  645 

.37  355 

.03  576 

.40  931 

4 

57 

.59  098 

.96  419 

.62  680 

.37  320 

.03  581 

.40  902 

3 

58 

.59  128 

.96  413 

.62  715 

.37  285 

.03  587 

.40  872 

2 

59 

.59  158 

.96  408 

.62  750 

.37  250 

.03  592 

.40842 

1 

60 

9.59  188 

9.96  403 

9.62  785 

0.37  215 

0.03  597 

0.40  812 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

112°  (292°) 


(247°)  67° 


Table  4.    Trigonometric  Logarithms 


219 


23°  (203°) 


(336°)  156° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.59  188 

9.96  403 

9.62  785 

0.37  215 

0.03  597 

0.40  812 

60 

1 

.59  218 

.96  397 

.62  820 

.37  180 

.03  603 

.40  782 

59 

2 

.59  247 

.96  392 

.62  855 

.37  145 

.03  608 

.40  753 

58 

3 

.59  277 

.96  387 

.62  890 

.37  110 

.03  613 

.40  723 

57 

4 

.59  307 

.96  381 

.62  926 

.37  074 

.03  619 

.40  693 

56 

5 

9.59  336 

9.96  376 

9.62  961 

0.37  039 

0.03  624 

0.40  664 

55 

6 

.59  366 

.96  370 

.62  996 

.37  004 

.03  630 

.40  634 

54 

7 

.59  396 

.96  365 

.63  031 

.36  969 

.03  635 

.40  604 

53 

8 

.59  425 

.96  360 

.63  066 

.36  934 

.03  640 

.40  575 

52 

9 

.59  455 

.96  354 

.63  101 

.36  899 

.03  646 

.40  545 

51 

10 

9.59  484 

9.96  349 

9.63  135 

0.36  865 

0.03  651 

0.40  516 

50 

11 

.59  514 

.96  343 

.63  170 

.36  830 

.03  657 

.40  486 

49 

12 

.59  543 

.96  338 

.63  205 

.36  795 

.03  662 

.40  457 

48 

13 

.59  573 

.96  333 

.63  240 

.36  760 

.03  667 

.40  427 

47 

14 

.59  602 

.96  327 

.63  275 

.36  725 

.03  673 

.40  398 

46 

15 

9.59  632 

9.96  322 

9.63  310 

0.36  690 

0.03  678 

0.40  368 

45 

16 

.59  661 

.96  316 

.63  345 

.36  655 

.03  684 

.40  339 

44 

17 

.59  690 

.96311 

.63  379 

.36  621 

.03  689 

.40  310 

43 

18 

.59  720 

.96  305 

.63  414 

.36  586 

.03  695 

.40  280 

42 

19 

.59  749 

.96  300 

.63  449 

.36  551 

.03  700 

.40  251 

41 

20 

9.59  778 

9.96  294 

9.63  484 

0.36  516 

0.03  706 

0.40  222 

40 

21 

.59  808 

.96  289 

.63  519 

.36  481 

.03  711 

.40  192 

39 

22 

.59  837 

.96284 

.63  553 

.36  447 

.03  716 

.40  163 

38 

23 

.59  866 

.96  278 

.63  588 

.36  412 

.03  722 

.40  134 

37 

24 

.59  895 

.96  273 

.63  623 

.36  377 

.03  727 

.40  105 

36 

25 

9.59  924 

9.96  267 

9.63  657 

0.36  343 

0.03  733 

0.40  076 

35 

26 

.59  954 

.96  262 

.63  692 

.36  308 

.03  738 

.40  046 

34 

27 

.59  983 

.96  256 

.63  726 

.36  274 

.03  744 

.40  017 

33 

28 

.60  012 

.96  251 

.63  761 

.36  239 

.03  749 

.39  988 

32 

29 

.60041 

.96  245 

.63  796 

.36  204 

.03  755 

.39  959 

31 

30 

9.60  070 

9.96  240 

9.63  830 

0.36  170 

0.03  760 

.39  930 

30 

31 

.60  099 

.96  234 

.63  865 

.36  135 

.03  766 

.39  901 

29 

32 

.60  128 

.96  229 

.63  899 

.36  101 

.03  771 

.39  872 

28 

33 

.60  157 

.96  223 

.63  934 

.36  066 

.03  777 

.39843 

27 

34 

.60  186 

.96  218 

.63  968 

.36  032 

.03  782 

.39  814 

26 

35 

9.60  215 

9.96  212 

9.64  003 

0.35  997 

0.03  788 

0.39  785 

25 

36 

.60244 

.96  207 

.64  037 

.35  963 

.03  793 

.39  756 

24 

37 

.60  273 

.96  201 

.64072 

.35  928 

.03  799 

.39  727 

23 

38 

.60  302 

.96  196 

.64  106 

.35  894 

.03  804 

.39  698 

22 

39 

.60  331 

.96  190 

.64  140 

.35  860 

.03  810 

.39  669 

21 

40 

9.60  359 

9.96  185 

9.64  175 

0.35  825 

0.03  815 

0.39  641 

20 

41 

.60  388 

.96  179 

.64  209 

.35  791 

.03  821 

.39  612 

19 

42 

.60  417 

.96  174 

.64243 

.35  757 

.03  826 

.39  583 

18 

43 

.60446 

.96  168 

.64278 

.35  722 

.03  832 

.39  554 

17 

44 

.60  474 

.96  162 

.64312 

.35  688 

.03  838 

.39  526 

16 

45 

9.60  503 

9.96  157 

9.64  346 

0.35  654 

0.03  843 

0.39  497 

15 

46 

.60  532 

.96  151 

.64  381 

.35  619 

.03849 

.39  468 

14 

47 

.60  561 

.96  146 

.64  415 

.35  585 

.03  854 

.39  439 

13 

48 

.60  589 

.96  140 

.64449 

.35  551 

.03  860 

.39411 

12 

49 

.60  618 

.96  135 

.64483 

.35  517 

.03  865 

.39  382 

11 

50 

9.60  646 

9.96  129 

9.64517 

0.35  483 

0.03  871 

0.39  354 

10 

51 

.60  675 

.96  123 

.64  552 

.35448 

.03  877 

.39  325 

9 

52 

.60  704 

.96  118 

.64586 

.35  414 

.03  882 

.39  296 

8 

53 

.60  732 

.96112 

.64620 

.35  380 

.03  888 

.39  268 

7 

54 

.60  761 

.96  107 

.64654 

.35  346 

.03  893 

.39  239 

6 

55 

9.60  789 

9.96  101 

9.64  688 

0.35  312 

0.03  899 

0.39  211 

5 

56 

.60  818 

.96  095 

.64  722 

.35  278 

.03  905 

.39  182 

4 

57 

.60846 

.96  090 

.64  756 

.35244 

.03  910 

.39  154 

3 

58 

.60  875 

.96084 

.64  790 

.35  210 

.03  916 

.39  125 

2 

59 

.60  903 

.96  079 

.64  824 

.35  176 

.03  921 

,39  097 

1 

60 

9.60  931 

9.96  073 

9.64  858 

0.35  142 

0.03  927 

0.39  069 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

113°  (293°) 


(246°)  66C 


220 


Table  4.    Trigonometric  Logarithms 


24°  (204°) 


(335°)  155' 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.60  931 

9.96  073 

9.64  858 

0.35  142 

0.03  927 

0.39  069 

60 

1 

.60  960 

.96  067 

.64  892 

.35  108 

.03  933 

.39  040 

59 

2 

.60  988 

.96  062 

.64926 

.35  074 

.03  938 

.39  012 

58 

3 

.61  016 

.96  056 

.64  960 

.35  040 

.03  944 

.38  984 

57 

4 

.61  045 

.96  050 

.64  994 

.35  006 

.03  950 

.38  955 

56 

5 

9.61  073 

9.96  045 

9.65  028 

0.34  972 

0.03  955 

0.38  927 

55 

6 

.61  101 

.96  039 

.65  062 

.34  938 

.03  961 

.38  899 

54 

7 

.61  129 

.96  034 

.65  096 

.34  904 

.03  966 

.38  871 

53 

8 

.61  158 

.96  028 

.65  130 

.34  870 

.03  972 

.38  842 

52 

9 

.61  186 

.96  022 

.65  164 

.34  836 

.03  978 

.38  814 

51 

10 

9.61  214 

9.96  017 

9.65  197 

0.34  803 

0.03  983 

0.38  786 

50 

11 

.61  242 

.96  Oil 

.65  231 

.34  769 

.03  989 

.38  758 

49 

12 

.61  270 

.96  005 

.65  265 

.34  735 

.03  995 

.38  730 

48 

13 

.61  298 

.96  000 

.65  299 

.34  701 

.04  000 

.38  702 

47 

14 

.61  326 

.95  994 

.65  333 

.34  667 

.04  006 

.38  674 

46 

15 

9.61  354 

9.95  988 

9.65  366 

0.34  634 

0.04  012 

0.38  646 

45 

16 

.61  382 

.95  982 

.65  400 

.34  600 

.04  018 

.38618 

44 

17 

.61  411 

.95  977 

.65  434 

.34  566 

.04  023 

.38  589 

43 

18 

.61  438 

.95  971 

.65  467 

.34  533 

.04  029 

.38  562 

42 

19 

.61  466 

.95  965 

.65  501 

.34  499 

.04  035 

.38  534 

41 

20 

9.61  494 

9.95  960 

9.65  535 

0.34  465 

0.04  040 

0.38  506 

40 

21 

.61  522 

.95  954 

.65  568 

.34  432 

.04  046 

.38  478 

39 

22 

.61  550 

.95  948 

.65  602 

.34  398 

.04  052 

.38  450 

38 

23 

.61  578 

.95  942 

.65  636 

.34  364 

.04  058 

.38  422 

37 

24 

.61  606 

.95  937 

.65  669 

.34  331 

.04  063 

.38  394 

36 

25 

9.61  634 

9.95  931 

9.65  703 

0.34  297 

0.04  069 

0.38  366 

35 

26 

.61  662 

.95  925 

.65  736 

.34  264 

.04  075 

.38  338 

34 

27 

.61  689 

.95  920 

.65  770 

.34  230 

.04  080 

.38311 

33 

28 

.61  717 

.95  914 

.65  803 

.34  197 

.04  086 

.38  283 

32 

29 

.61  745 

.95  908 

.65  837 

.34  163 

.04  092 

.38  255 

31 

30 

9.61  773 

9.95  902 

9.65  870 

0.34  130 

0.04  098 

0.38  227 

30 

31 

.61  800 

.95  897 

.65  904 

.34  096 

.04  103 

.38  200 

29 

32 

.61  828 

.95  891 

.65  937 

.34  063 

.04  109 

.38  172 

28 

33 

.61  856 

.95  885 

.65  971 

.34  029 

.04  115 

.38  144 

27 

34 

.61  883 

.95  879 

.66  004 

.33  996 

.04  121 

.38  117 

26 

35 

9.61  911 

9.95  873 

9.66  038 

0.33  962 

0.04  127 

0.38  089 

25 

36 

.61  939 

.95  868 

.66  071 

.33  929 

.04  132 

.38  061 

24 

37 

.61  966 

.95  862 

.66  104 

.33  896 

.04  138 

.38  034 

23 

38 

.61  994 

.95  856 

.66  138 

.33  862 

.04  144 

.38  006 

22 

39 

.62  021 

.95  850 

.66  171 

.33  829 

.04  150 

.37  979 

21 

40 

9.62  049 

9.95  844 

9.66  204 

0.33  796 

0.04  156 

0.37  951 

20 

41 

.62  076 

.95  839 

.66  238 

.33  762 

.04  161 

.37  924 

19 

42 

.62  104 

.95  833 

.66  271 

.33  729 

.04  167 

.37  896 

18 

43 

.62  131 

.95  827 

.66  304 

.33  696 

.04  173 

.37  869 

17 

44 

.62  159 

.95  821 

.66  337 

.33  663 

.04  179 

.37841 

16 

45 

9.62  186 

9.95  815 

9.66  371 

0.33  629 

0.04  185 

0.37  814 

15 

46 

.62  214 

.95  810 

.66  404 

.33  596 

.04  190 

.37  786 

14 

47 

.62  241 

.95  804 

.66  437 

.33  563 

.04  196 

.37  759 

13 

48 

.62  268 

.95  798 

.66  470 

.33  530 

.04  202 

.37  732 

12 

49 

.62  296 

.95  792 

.66  503 

.33  497 

.04  208 

.37  704 

11 

50 

9.62  323 

9.95  786 

9.66  537 

0.33  463 

0.04  214 

0.37  677 

10 

51 

.62  350 

.95  780 

.66  570 

.33  430 

.04  220 

.37  650 

9 

52 

.62  377 

.95  775 

.66  603 

.33  397 

.04  225 

.37  623 

8 

53 

.62  405 

.95  769 

.66  636 

.33  364 

.04  231 

.37  595 

7 

54 

.62  432 

.95  763 

.66  669 

.33  331 

.04  237 

.37  568 

6 

55 

9.62  459 

9.95  757 

9.66  702 

0.33  298 

0.04  243 

0.37  541 

5 

56 

.62  486 

.95  751 

.66  735 

.33  265 

.04  249 

.37  514 

4 

57 

.62  513 

.95  745 

.66  768 

.33  232 

.04  255 

.37  487 

3 

58 

.62  541 

.95  739 

.66  801 

.33  199 

.04  261 

.37  459 

2 

59 

.62  568 

.95  733 

.66  834 

.33  166 

.04  267 

.37  432 

1 

60 

9.62  595 

9.95  728 

9.66  867 

0.33  133 

0.04  272 

0.37  405 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

114°  (294°) 


(245°)  65C 


Table  4.    Trigonometric  Logarithms 


221 


25°  (205°) 


(334°)  154C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.62  595 

9.95  728 

9.66  867 

0.33  133 

0.04  272 

0.37  405 

60 

1 

.62  622 

.95  722 

.66  900 

.33  100 

.04  278 

.37  378 

59 

2 

.62  649 

.95  716 

.66  933 

.33  067 

.04284 

.37  351 

58 

3 

.62  676 

.95  710 

.66  966 

.33  034 

.04  290 

.37  324 

57 

4 

.62  703 

.95  704 

.66999 

.33  001 

.04  296 

.37  297 

56 

5 

9.62  730 

9.95  698 

9.67  032 

0.32  968 

0.04  302 

0.37  270 

55 

6 

.62  757 

.95  692 

.67  065 

.32  935 

.04308 

.37  243 

54 

7 

.62784 

.95  686 

.67  098 

.32  902 

.04  314 

.37  216 

53 

8 

.62811 

.95  680 

.67  131 

.32  869 

.04  320 

.37  189 

52 

9 

.62838 

.95  674 

.67  163 

.32  837 

.04  326 

.37  162 

51 

10 

9.62  865 

9.95  668 

9.67  196 

0.32  804 

0.04  332 

0.37  135 

50 

11 

.62  892 

.95  663 

.67  229 

.32  771 

.04337 

.37  108 

49 

12 

.62  918 

.95  657 

.67  262 

.32  738 

.04  343 

.37  082 

48 

13 

.62  945 

.95  651 

.67  295 

.32  705 

.04  349 

.37  055 

47 

14 

.62  972 

.95645 

.67  327 

.32  673 

.04355 

.37  028 

46 

15 

9.62  999 

9.95  639 

9.67  360 

0.32  640 

0.04  361 

0.37  001 

45 

16 

.63  026 

.95  633 

.67  393 

.32  607 

.04  367 

.36  974 

44 

17 

.63  052 

.95  627 

.67  426 

.32  574 

.04  373 

.36  948 

43 

18 

.63  079 

.95  621 

.67  458 

.32  542 

.04  379 

.36  921 

42 

19 

.63  106 

.95  615 

.67  491 

.32  509 

.04385 

.36  894 

41 

20 

9.63  133 

9.95  609 

9.67  524 

0.32  476 

0.04  391 

0.36  867 

40 

21 

.63  159 

.95  603 

.67  556 

.32  444 

.04  397 

.36841 

39 

22 

.63  186 

.95  597 

.67  589 

.32411 

.04  403 

.36  814 

38 

23 

.63  213 

.95  591 

.67  622 

.32  378 

.04409 

.36  787 

37 

24 

.63  239 

.95  585 

.67  654 

.32  346 

.04  415 

.36  761 

36 

25 

9.63  266 

9.95  579 

9.67  687 

0.32  313 

0.04  421 

0.36  734 

35 

26 

.63  292 

.95  573 

.67  719 

.32  281 

.04  427 

.36  708 

34 

27 

.63  319 

.95  567 

.67  752 

.32  248 

.04  433 

.36  681 

33 

28 

.63  345 

.95  561 

.67  785 

.32  215 

.04  439 

.36  655 

32 

29 

.63  372 

.95  555 

.67  817 

.32  183 

.04  445 

.36  628 

31 

30 

9.63  398 

9.95  549 

9.67  850 

0.32  150 

0.04  451 

0.36  602 

30 

31 

.63  425 

.95  543 

.67  882 

.32  118 

.04  457 

.36  575 

29 

32 

.63  451 

.95  537 

.67  915 

.32  085 

.04  463 

.36  549 

28 

33 

.63  478 

.95  531 

.67  947 

.32  053 

.04  469 

.36  522 

27 

34 

.63  504 

.95  525 

.67  980 

.32  020 

.04  475 

.36  496 

26 

35 

9.63  531 

9.95  519 

9.68012 

0.31  988 

0.04  481 

0.36  469 

25 

36 

.63  557 

.95  513 

.68  044 

.31  956 

.04  487 

.36  443 

24 

37 

.63583 

.95  507 

.68  077 

.31  923 

.04  493 

.36  417 

23 

38 

.63  610 

.95  500 

.68  109 

.31  891 

.04  500 

.36  390 

22 

39 

.63  636 

.95  494 

.68  142 

.31  858 

.04506 

.36  364 

21 

40 

9.63  662 

9.95  488 

9.68  174 

0.31  826 

0.04  512 

0.36  338 

20 

41 

.63  689 

.95  482 

.68  206 

.31  794 

.04  518 

.  36311 

19 

42 

.63  715 

.95  476 

.68  239 

.31  761 

.04  524 

.36  285 

18 

43 

.63  741 

.95  470 

.68  271 

.31  729 

.04  530 

.36  259 

17 

44 

.63  767 

.95  464 

.68  303 

.31  697 

.04536 

.36  233 

16 

45 

9.63  794 

9.95  458 

9.68  336 

0.31  664 

0.04  542 

0.36  206 

15 

46 

.63  820 

.95  452 

.68  368 

.31  632 

.04  548 

.36  180 

14 

47 

.63846 

.95  446 

.68  400 

.31  600 

.04554 

.36  154 

13 

48 

.63  872 

.95  440 

.68  432 

.31  568 

.04  560 

.36  128 

12 

49 

.63  898 

.95  434 

.68  465 

.31  535 

.04  566 

.36  102 

11 

50 

9.63  924 

9.95  427 

9.68  497 

0.31  503 

0.04  573 

0.36  076 

10 

51 

.63  950 

.95  421 

.68  529 

.31  471 

.04  579 

.36  050 

9 

52 

.63  976 

.95  415 

.68  561 

.31  439 

.04  585 

.36  024 

8 

53 

.64  002 

.95  409 

.68  593 

.31  407 

.04  591 

.35  998 

7 

54 

.64  028 

.95  403 

.68  626 

.31  374 

.04597 

.35  972 

6 

55 

9.64  054 

9.95  397 

9.68  658 

0.31  342 

0.04  603 

0.35  946 

5 

56 

.64080 

.95  391 

.68  690 

.31  310 

.04  609 

.35  920 

4 

57 

.64  106 

.95  384 

.68  722 

.31  278 

.04  616 

.35  894 

3 

58 

.64132 

.95  378 

.68  754 

.31  246 

.04  622 

.35  868 

2 

59 

.64158 

.95  372 

.68  786 

.31  214 

.04  628 

.35  842 

1 

60 

9.64  184 

9.95  366 

9.68  818 

0.31  182 

0.04  634 

0.35  816 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

115°  (295°) 


(244°)  64' 


222 


Table  4.    Trigonometric  Logarithms 


26°  (206°) 


(333°)  153C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.64  184 

9.95  366 

9.68818 

0.31  182 

0.04  634 

0.35  816 

60 

1 

.64  210 

.95  360 

.68  850 

.31  150 

.04  640 

.35  790 

59 

2 

.64  236 

.95  354 

.68  882 

.31  118 

.04  646 

.35  764 

58 

3 

.64  262 

.95  348 

.68  914 

.31  086 

.04  652 

.35  738 

57 

4 

.64  288 

.95  341 

.68  946 

.31  054 

.04  659 

.35  712 

56 

5 

9.64  313 

9.95  335 

9.68  978 

0.31  022 

0.04  665 

0.35  687 

55 

6 

.64339 

.95  329 

.69  010 

.30990 

.04  671 

.35  661 

54 

7 

.64  365 

.95  323 

.69  042 

.30  958 

.04677 

.35  635 

53 

8 

.64  391 

.95  317 

.69  074 

.30  926 

.04  683 

.35  609 

52 

9 

.64  417 

.95  310 

.69  106 

.30  894 

.04  690 

.35  583 

51 

10 

9.64  442 

9.95  304 

9.69  138 

0.30  862 

0.04  696 

0.35  558 

50 

11 

.64  468 

.95  298 

.69  170 

.30  830 

.04  702 

.35  532 

49 

12 

.64  494 

.95  292 

.69  202 

.30  798 

.04  708 

.35  506 

48 

13 

.64  519 

.95  286 

.69  234 

.30  766 

.04  714 

.35  481 

47 

14 

.64  545 

.95  279 

.69  266 

.30  734 

.04  721 

.35  455 

46 

15 

9.64  571 

9.95  273 

9.69  298 

0.30  702 

0.04  727 

0.35  429 

45 

16 

.64  596 

.95  267 

.69  329 

.30  671 

.04  733 

.35  404 

44 

17 

.64  622 

.95  261 

.69  361 

.30  639 

.04  739 

.35  378 

43 

18 

.64  647 

.95  254 

.69  393 

.30  607 

.04  746 

.35353 

42 

19 

.64  673 

.95  248 

.69  425 

.30  575 

.04  752 

.35  327 

41 

20 

9.64  698 

9.95  242 

9.69  457 

0.30  543 

0.04  758 

0.35  302 

40 

21 

.64  724 

.95  236 

.69  488 

.30  512 

.04  764 

.35  276 

39 

22 

.64  749 

.95  229 

.69  520 

.30  480 

.04  771 

.35  251 

38 

23 

.64  775 

.95  223 

.69  552 

.30  448 

.04  777 

.35  225 

37 

24 

.64  800 

.95  217 

.69  584 

.30  416 

.04  783 

.35  200 

36 

25 

9.64  826 

9.95211 

9.69  615 

0.30  385 

0.04  789 

0.35  174 

35 

26 

.64  851 

.95  204 

.69  647 

.30  353 

.04  796 

.35  149 

34 

27 

.64877 

.95  198 

.69  679 

.30  321 

.04  802 

.35  123 

33 

28 

.64  902 

.95  192 

.69  710 

.30  290 

.04808 

.35  098 

32 

29 

.64  927 

.95  185 

.69  742 

.30  258 

.04  815 

.35  073 

31 

30 

9.64  953 

9.95  179 

9.69  774 

0.30  226 

0.04  821 

0.35  047 

30 

31 

.64  978 

.95  173 

.69  805 

.30  195 

.04  827 

.35  022 

29 

32 

.65  003 

.95  167 

.69  837 

.30  163 

.04  833 

.34  997 

28 

33 

.65  029 

.95  160 

.69  868 

.30  132 

.04  840 

.34  971 

27 

34 

.65  054 

.95  154 

.69  900 

.30  100 

.04846 

.34  946 

26 

35 

9.65  079 

9.95  148 

9.69  932 

0.30  068 

0.04  852 

0.34  921 

25 

36 

.65  104 

.95  141 

.69  963 

.30  037 

.04  859 

.34  896 

24 

37 

.65  130 

.95  135 

.69  995 

.30  005 

.04  865 

.34  870 

23 

38 

.65  155 

.95  129 

.70  026 

.29  974 

.04  871 

.34  845 

22 

39 

.65  180 

.95  122 

.70  058 

.29  942 

.04  878 

.34  820 

21 

40 

9.65  205 

9.95  116 

9.70  089 

0.29911 

0.04  884 

0.34  795 

20 

41 

.65  230 

.95  110 

.70  121 

.29  879 

.04  890 

.34  770 

19 

42 

.65  255 

.95  103 

.70  152 

.29848 

.04  897 

.34  745 

18 

43 

.65  281 

.95  097 

.70  184 

.29  816 

.04  903 

.34  719 

17 

44 

.65  306 

.95  090 

.70  215 

.29  785 

.04  910 

.34  694 

16 

45 

9.65  331 

9.95  084 

9.70  247 

0.29  753 

0.04  916 

0.34  669 

15 

46 

.65  356 

.95  078 

.70  278 

.29  722 

.04  922 

.34  644 

14 

47 

.65  381 

.95  071 

.70  309 

.29  691 

.04  929 

.34  619 

13 

48 

.65  406 

.95  065 

.70  341 

.29  659 

.04  935 

.34  594 

12 

49 

.65  431 

.95  059 

.70  372 

.29  628 

.04  941 

.34  569 

11 

50 

9.65  456 

9.95  052 

9.70  404 

0.29  596 

0.04  948 

0.34  544 

10 

51 

.65  481 

.95  046 

.70  435 

.29  565 

.04  954 

.34  519 

9 

52 

.65  506 

.95  039 

.70  466 

.29  534 

.04  961 

.34  494 

8 

53 

.65  531 

.95  033 

.70  498 

.29  502 

.04  967 

.34  469 

7 

54 

.65  556 

.95  027 

.70  529 

.29  471 

.04  973 

.34  444 

6 

55 

9.65  580 

9.95  020 

9.70  560 

0.29  440 

0.04  980 

0.34  420 

5 

56 

.65  605 

.95  014 

.70  592 

.29  408 

.04  986 

.34  395 

4 

57 

.65  630 

.95  007 

.70  623 

.29  377 

.04  993 

.34  370 

3 

58 

.65  655 

.95  001 

.70  654 

.29  346 

.04  999 

.34  345 

2 

59 

.65  680 

.94  995 

.70  685 

.29  315 

.05  005 

.34  320 

1 

60 

9.65  705 

9.94  988 

9.70  717 

0.29  283 

0.05  012 

0.34  295 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

116°  (296°) 


(243°)  63C 


Table  4.    Trigonometric  Logarithms 


223 


27°  (207°) 


(332°)  152° 


Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.65  705 

9.94  988 

9.70717 

0.29  283 

0.05  012 

0.34  295 

60 

1 

.65  729 

.94  982 

.70  748 

.29  252 

.05  018 

.34  271 

59 

2 

.65  754 

.94  975 

.70  779 

.29  221 

.05  025 

.34  246 

58 

3 

.65  779 

.94  969 

.70  810 

.29  190 

.05  031 

.34  221 

57 

4 

.65  804 

.94  962 

.70841 

.29  159 

.05  038 

.34  196 

56 

5 

9.65  828 

9.94  956 

9.70  873 

0.29  127 

0.05  044 

0.34  172 

55 

6 

.65  853 

.94  949 

.70  904 

.29  096 

.05  051 

.34  147 

54 

7 

.65  878 

.94  943 

.70  935 

.29  065 

.05  057 

.34  122 

53 

8 

.65  902 

.94  936 

.70  966 

.29  034 

.05  064 

.34  098 

52 

9 

.65  927 

.94  930 

.70  997 

.29  003 

.05  070 

.34  073 

51 

10 

9.65  952 

9.94  923 

9.71  028 

0,28  972 

0.05  077 

0.34  048 

50 

11 

.65  976 

.94  917 

.71  059 

.28  941 

.05  083 

.34  024 

49 

12 

.66  001 

.94911 

.71  090 

.28  910 

.05  089 

.33  999 

48 

13 

.66  025 

.94  904 

.71  121 

.28  879 

.05  096 

.33  975 

47 

14 

.66  050 

.94  898 

.71  153 

.28847 

.05  102 

.33  950 

46 

15 

9.66  075 

9.94  891 

9.71  184 

0.28  816 

0.05  109 

0.33  925 

45 

16 

.66  099 

.94  885 

.71  215 

.28  785 

.05  115 

.33  901 

44 

17 

.66  124 

.94  878 

.71  246 

.28  754 

.05  122 

.33  876 

43 

18 

.66  148 

.94  871 

.71  277 

.28  723 

.05  129 

.33  852 

42 

19 

.66  173 

.94  865 

.71  308 

.28  692 

.05  135 

.33  827 

41 

20 

9.66  197 

9.94  858 

9.71  339 

0.28  661 

0.05  142 

0.33  803 

40 

21 

.66  221 

.94  852 

.71  370 

.28  630 

.05  148 

.33  779 

39 

22 

.66  246 

.94845 

.71  401 

.28  599 

.05  155 

.33  754 

38 

23 

.66  270 

.94  839 

.71  431 

.28  569 

.05  161 

.33  730 

37 

24 

.66  295 

.94  832 

.71  462 

.28  538 

.05  168 

.33  705 

36 

25 

9.66  319 

9.94  826 

9.71  493 

0.28  507 

0.05  174 

0.33  681 

35 

26 

.66  343 

.94  819 

.71  524 

.28  476 

.05  181 

.33  657 

34 

27 

.66  368 

.94  813 

.71  555 

.28  445 

.05  187 

.33  632 

33 

28 

.66  392 

.94  806 

.71  586 

.28  414 

.05  194 

.33  608 

32 

29 

.66  416 

.94  799 

.71  617 

.28  383 

.05  201 

.33  584 

31 

30 

9.66  441 

9.94  793 

9.71  648 

0.28  352 

0.05  207 

0.33  559 

30 

31 

.66  465 

.94  786 

.71  679 

.28  321 

.05  214 

.33  535 

29 

32 

.66  489 

.94  780 

.71  709 

.28  291 

.05  220 

.33511 

28 

33 

.66  513 

.94  773 

.71  740 

.28  260 

.05  227 

.33  487 

27 

34 

.66  537 

.94  767 

.71  771 

.28  229 

.05  233 

.33  463 

26 

35 

9.66  562 

9.94  760 

9.71  802 

0.28  198 

0.05  240 

0.33  438 

25 

36 

.66  586 

.94  753 

.71  833 

.28  167 

.05  247 

.33  414 

24 

37 

.66  610 

.94  747 

.71  863 

.28  137 

.05  253 

.33  390 

23 

38 

.66  634 

.94  740 

.71  894 

.28  106 

.05  260 

.33  366 

22 

39 

.66  658 

.94  734 

.71  925 

.28  075 

.05  266 

.33  342 

21 

40 

9.66  682 

9.94  727 

9.71  955 

0.28  045 

0.05  273 

0.33  318 

20 

41 

.66  706 

.94  720 

.71  986 

.28  014 

.05  280 

.33  294 

19 

42 

.66  731 

.94  714 

.72  017 

.27  983 

.05  286 

.33  269 

18 

43 

.66  755 

.94  707 

.72  048 

.27  952 

.05  293 

.33  245 

17 

44 

.66  779 

.94  700 

.72  078 

.27  922 

.05  300 

.33  221 

16 

45 

9.66  803 

9.94  694 

9.72  109 

0.27  891 

0.05  306 

0.33  197 

15 

46 

.66  827 

.94  687 

.72  140 

.27  860 

.05  313 

.33  173 

14 

47 

.66  851 

.94  680 

.72  170 

.27  830 

.05  320 

.33  149 

13 

48 

.66  875 

.94  674 

.72  201 

.27  799 

.05  326 

.33  125 

12 

49 

.66  899 

.94667 

.72  231 

.27  769 

.05  333 

.33  101 

11 

50 

9.66  922 

9.94  660 

9.72  262 

0.27  738 

0.05  340 

0.33  078 

10 

51 

.66  946 

.94  654 

.72  293 

.27  707 

.05  346 

.33  054 

9 

52 

.66  970 

.94647 

.72  323 

.27  677 

.05  353 

.33  030 

8 

53 

.66  994 

.94  640 

.72  354 

.27646 

.05  360 

.33  006 

7 

54 

.67  018 

.94  634 

.72384 

.27  616 

.05  366 

.32  982 

6 

55 

9.67  042 

9.94  627 

9.72  415 

0.27  585 

0.05  373 

0.32  958 

5 

56 

.67  066 

.94  620 

.72445 

.27  555 

.05  380 

.32  934 

4 

57 

.67  090 

.94  614 

.72  476 

.27  524 

.05  386 

.32  910 

3 

58 

.67  113 

.94  607 

.72  506 

.27  494 

.05  393 

.32  887 

2 

59 

.67  137 

.94  600 

.72  537 

.27  463 

.05400 

.32  863 

1 

60 

9.67  161 

9.94  593 

9.72  567 

0.27  433 

0.05  407 

0.32  839 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

117°  (297°) 


(242°)  62° 


224 


Table  4.    Trigonometric  Logarithms 


28°  (208°) 


(331°)  151C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.67  161 

9.94  593 

9.72  567 

0.27  433 

0.05  407 

0.32  839 

60 

1 

.67  185 

.94  587 

.72  598 

.27  402 

.05413 

.32  815 

59 

2 

.67  208 

.94  580 

.72  628 

.27  372 

.05  420 

.32  792 

58 

3 

.67  232 

.94  573 

.72  659 

.27  341 

.05  427 

.32  768 

57 

4 

.67  256 

.94  567 

.72  689 

.27311 

.05  433 

.32  744 

56 

5 

9.67  280 

9.94  560 

9.72  720 

0.27  280 

0.05  440 

0.32  720 

55 

6 

.67  303 

.94  553 

.72  750 

.27  250 

.05  447 

.32  697 

54 

7 

.67  327 

.94  546 

.72  780 

.27  220 

.05  454 

.32  673 

53 

8 

.67  350 

.94  540 

.72811 

.27  189 

.05  460 

.32  650 

52 

9 

.67  374 

.94  533 

.72841 

.27  159 

.05  467 

.32  626 

51 

10 

9.67  398 

9.94  526 

9.72  872 

0.27  128 

0.05  474 

0.32  602 

50 

11 

.67  421 

.94  519 

.72  902 

.27  098 

.05  481 

.32  579 

49 

12 

.67  445 

.94  513 

.72  932 

.27  068 

.05  487 

.32  555 

48 

13 

.67  468 

.94  506 

.72  963 

.27  037 

.05  494 

.32  532 

47 

14 

.67  492 

.94  499 

.72  993 

.27  007 

.05  501 

.32  508 

46 

15 

9.67  515 

9.94  492 

9.73  023 

0.26  977 

0.05  508 

0.32  485 

45 

16 

.67  539 

.94  485 

.73  054 

.26  946 

.05  515 

.32  461 

44 

17 

.67  562 

.94  479 

.73084 

.26  916 

.05  521 

.32  438 

43 

18 

.67  586 

.94  472 

.73  114 

.26  886 

.05  528 

.32  414 

42 

19 

.67  609 

.94  465 

.73  144 

.26  856 

.05  535 

.32  391 

41 

20 

9.67  633 

9.94  458 

9.73  175 

0.26  825 

0.05  542 

0.32  367 

40 

21 

.67  656 

.94  451 

.73  205 

.26  795 

.05  549 

.32  344 

39 

22 

.67  680 

.94  445 

.73  235 

.26  765 

.05  555 

.32  320 

38 

23 

.67  703 

.94  438 

.73  265 

.26  735 

.05  562 

.32  297 

37 

24 

.67  726 

.94  431 

.73  295 

.26  705 

.05  569 

.32  274 

36 

25 

9.67  750 

9.94  424 

9.73  326 

0.26  674 

0.05  576 

0.32  250 

35 

26 

.67  773 

.94  417 

.73  356 

.26  644 

.05  583 

.32  227 

34 

27 

.67  796 

.94  410 

.73  386 

.26  614 

.05  590 

.32  204 

33 

28 

.67  820 

.94  404 

.73  416 

.26  584 

.05  596 

.32  180 

32 

29 

.67  843 

.94  397 

.73  446 

.26  554 

.05  603 

.32  157 

31 

30 

9.67  866 

9.94  390 

9.73  476 

0.26  524 

0.05  610 

0.32  134 

30 

31 

.67  890 

.94  383 

.73  507 

.26  493 

.05  617 

.32  110 

29 

32 

.67  913 

.94  376 

.73  537 

.26  463 

.05  624 

.32  087 

28 

33 

.67  936 

.94  369 

.73  567 

.26  433 

.05  631 

.32  064 

27 

34 

.67  959 

.94  362 

.73  597 

.26  403 

.05  638 

.32  041 

26 

35 

9.67  982 

9.94  355 

9.73  627 

0.26  373 

0.05  645 

0.32  018 

25 

36 

.68  006 

.94  349 

.73  657 

.26  343 

.05  651 

.31  994 

24 

37 

.68  029 

.94  342 

.73  687 

.26  313 

.05  658 

.31  971 

23 

38 

.68  052 

.94  335 

.73  717 

.26  283 

.05  665 

.31  948 

22 

39 

.68  075 

.94  328 

.73  747 

.26  253 

.05  672 

.31  925 

21 

40 

9.68  098 

9.94  321 

9.73  777 

0.26  223 

0.05  679 

0.31  902 

20 

41 

.68  121 

.94  314 

.73  807 

.26  193 

.05  686 

.31  879 

19 

42 

.68  144 

.94  307 

.73  837 

.26  163 

.05  693 

.31  856 

18 

43 

.68  167 

.94  300 

.73  867 

.26  133 

.05  700 

.31  833 

17 

44 

.68  190 

.94  293 

.73  897 

.26  103 

.05  707 

.31  810 

16 

45 

9.68  213 

9.94  286 

9.73  927 

0.26  073 

0.05  714 

0.31  787 

15 

46 

.68  237 

.94  279 

.73  957 

.26  043 

.05  721 

.31  763 

14 

47 

.68  260 

.94  273 

.73  987 

.26  013 

.05  727 

.31  740 

13 

48 

.68  283 

.94  266 

.74  017 

.25  983 

.05  734 

.31  717 

12 

49 

.68  305 

.94  259 

.74  047 

.25  953 

.05  741 

.31  695 

11 

50 

9.68  328 

9.94  252 

9.74  077 

0.25  923 

0.05  748 

0.31  672 

10 

51 

.68  351 

.94  245 

.74  107 

.25  893 

.05  755 

.31  649 

9 

52 

.68  374 

.94  238 

.74  137 

.25  863 

.05  762 

.31  626 

8 

53 

.68  397 

.94  231 

.74  166 

.25  834 

.05  769 

.31  603 

7 

54 

.68  420 

.94  224 

.74  196 

.25  804 

.05  776 

.31  580 

6 

55 

9.68  443 

9.94  217 

9.74  226 

0.25  774 

0.05  783 

0.31  557 

5 

56 

.68  466 

.94  210 

.74  256 

.25  744 

.05  790 

.31  534 

4 

57 

.68  489 

.94  203 

.74  286 

.25  714 

.05  797 

.31  511 

3 

58 

.68512 

.94  196 

.74  316 

.25  684 

.05  804 

.31  488 

2 

59 

.68  534 

.94  189 

.74  345 

.25  655 

.05811 

.31  466 

1 

60 

9.68  557 

9.94  182 

9.74  375 

0.25  625 

0.05  818 

0.31  443 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

118°  (298°) 


(241°)  61' 


Table  4.    Trigonometric  Logarithms 


225 


29°  (209°) 


(330°)  150° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.68  557 

9.94  182 

9.74  375 

0.25  625 

0.05  818 

0.31  443 

60 

1 

.68  580 

.94  175 

.74  405 

.25  595 

.05  825 

.31  420 

59 

2 

.68  603 

.94  168 

.74  435 

.25  565 

.05  832 

.31  397 

58 

3 

.68  625 

.94  161 

.74  465 

.25535 

.05  839 

.31  375 

57 

4 

.68  648 

.94  154 

.74  494 

.25  506 

.05846 

.31  352 

56 

5 

9.68  671 

9.94  147 

9.74  524 

0.25  476 

0.05  853 

0.31  329 

55 

•   6 

.68  694 

.94  140 

.74  554 

.25  446 

.05  860 

.31  306 

54 

7 

.68  716 

.94  133 

.74  583 

.25417 

.05  867 

.31284 

53 

8 

.68  739 

.94  126 

.74  613 

.25  387 

.05  874 

.31  261 

52 

9 

.68  762 

.94  119 

.74643 

.25  357 

.05  881 

.31  238 

51 

10 

9.68  784 

9.94  112 

9.74  673 

0.25  327 

0.05  888 

0.31  216 

50 

11 

.68  807 

.94  105 

.74  702 

.25  298 

.05  895 

.31  193 

49 

12 

.68  829 

.94  098 

.74  732 

.25  268 

.05  902 

.31  171 

48 

13 

.68  852 

.94  090 

.74  762 

.25  238 

.05  910 

.31  148 

47 

14 

.68  875 

.94  083 

.74  791 

.25  209 

.05  917 

.31  125 

46 

15 

9.68  897 

9.94  076 

9.74  821 

0.25  179 

0.05  924 

0.31  103 

45 

16 

.68  920 

.94  069 

.74  851 

.25  149 

.05  931 

.31  080 

44 

17 

.68  942 

.94  062 

.74  880 

.25  120 

.05  938 

.31  058 

43 

18 

.68  965 

.94  055 

.74  910 

.25  090 

.05  945 

.31  035 

42 

19 

.68  987 

.94  048 

.74  939 

.25  061 

.05  952 

.31  013 

41 

20 

9.69  010 

9.94  041 

9.74  969 

0.25  031 

0.05  959 

0.30  990 

40 

21 

.69  032 

.94  034 

.74  998 

.25  002 

.05  966 

.30  968 

39 

22 

.69  055 

.94  027 

.75  028 

.24  972 

.05  973 

.30  945 

38 

23 

.69  077 

.94  020 

.75  058 

.24  942 

.05  980 

.30  923 

37 

24 

.69  100 

.94  012 

.75  087 

.24  913 

.05  988 

.30  900 

36 

25 

9.69  122 

9.94  005 

9.75  117 

0.24  883 

0.05  995 

0.30  878 

35 

26 

.69  144 

.93  998 

.75  146 

.24  854 

.06  002 

.30  856 

34 

27 

.69  167 

.93  991 

.75  176 

.24  824 

.06  009 

.30  833 

33 

28 

.69  189 

.93  984 

.75  205 

.24  795 

.06  016 

.30  811 

32 

29 

.69  212 

.93  977 

.75  235 

.24  765 

.06  023 

.30  788 

31 

30 

9.69  234 

9.93  970 

9.75  264 

0.24  736 

0.06  030 

0.30  766 

30 

31 

.69  256 

.93  963 

.75  294 

.24  706 

.06  037 

.30  744 

29 

32 

.69  279 

.93  955 

.75  323 

.24  677 

.06  045 

.30  721 

28 

33 

.69  301 

.93  948 

.75  353 

.24647 

.06  052 

.30  699 

27 

34 

.69  323 

.93  941 

.75  382 

.24  618 

.06059 

.30  677 

26 

35 

9.69  345 

9.93  934 

9.75411 

0.24  589 

0.06  066 

0.30  655 

25 

36 

.69  368 

.93  927 

.75  441 

.24  559 

.06  073 

.30  632 

24 

37 

.69  390 

.93  920 

.75  470 

.24  530 

.06  080 

.301610 

23 

38 

.69  412 

.93  912 

.75  500 

.24  500 

.06  088 

.30  588 

22 

39 

.69  434 

.93  905 

.75  529 

.24  471 

.06  095 

.30  566 

21 

40 

9.69  456 

9.93  898 

9.75  558 

0.24  442 

0.06  102 

0.30  544 

20 

41 

.69  479 

.93  891 

.75  588 

.24  412 

.06  109 

.30  521 

19 

42 

.69  501 

.93884 

.75  617 

.24  383 

.06  116 

.30  499 

18 

43 

.69  523 

.93  876 

.75647 

.24  353 

.06  124 

.30  477 

17 

44 

.69  545 

.93  869 

.75  676 

.24  324 

.06  131 

.30  455 

16 

45 

9.69  567 

9.93  862 

9.75  705 

0.24  295 

0.06  138 

0.30  433 

15 

46 

.69  589 

.93855 

.75  735 

.24  265 

.06  145 

.30411 

14 

47 

.69611 

.93847 

.75764 

.24  236 

.06  153 

.30  389 

13 

48 

.69  633 

.93840 

.75  793 

.24  207 

.06  160 

.30  367 

12 

49 

.69  655 

.93833 

.75  822 

.24  178 

.06167 

.30  345 

11 

50 

9.69  677 

9.93  826 

9.75  852 

0.24  148 

0.06  174 

0.30  323 

10 

51 

.69  699 

.93  819 

.75  881 

.24  119 

.06  181 

.30  301 

9 

52 

.69  721 

.93811 

.75  910 

.24  090 

.06  189 

.30  279 

8 

53 

.69  743 

.93804 

.75  939 

.24  061 

.06  196 

.30  257 

7 

54 

.69  765 

.93  797 

.75  969 

.24  031 

.06  203 

.30  235 

6 

55 

9.69  787 

9.93  789 

9.75  998 

0.24  002 

0.06211 

0.30213 

5 

56 

.69  809 

.93  782 

.76  027 

.23  973 

.06  218 

.30  191 

4 

57 

.69  831 

.93  775 

.76  056 

.23  944 

.06  225 

.30  169 

3 

58 

.69  853 

.93  768 

.76  086 

.23  914 

.06  232 

.30  147 

2 

59 

.69  875 

.93  760 

.76  115 

.23  885 

.06  240 

.30  125 

1 

60 

9.69  897 

9.93  753 

9.76  144 

0.23  856 

0.06  247 

0.30  103 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

119°  (299°) 


(240°)  60° 


226 


Table  4.    Trigonometric  Logarithms 


30°  (210°) 


(329°)  149° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.69  897 

9.93  753 

9.76  144 

0.23  856 

0.06  247 

0.30  103 

60 

1 

.69  919 

.93  746 

.76  173 

.23  827 

.06  254 

.30  081 

59 

2 

.69  941 

.93  738 

.76  202 

.23  798 

.06  262 

.30  059 

58 

3 

.69  963 

.93  731 

.76  231 

.23  769 

.06  269 

.30  037 

57 

4 

.69  984 

.93  724 

.76  261 

.23  739 

.06  276 

.30  016 

56 

5 

9.70  006 

9.93  717 

9.76  290 

0.23  710 

0.06  283 

0.29  994 

55 

6 

.70  028 

.93  709 

.76319 

.23  681 

.06  291 

.29  972 

54 

7 

.70  050 

.93  702 

.76  348 

.23  652 

.06  298 

.29  950 

53 

8 

.70  072 

.93  695 

.76  377 

.23  623 

.06  305 

.29  928 

52 

9 

.70  093 

.93  687 

.76  406 

.23  594 

.06  313 

.29  907 

51 

10 

9.70  115 

9.93  680 

9.76  435 

0.23  565 

0.06  320 

0.29  885 

50 

11 

.70  137 

.93  673 

.76  464 

.23  536 

.06  327 

.29  863 

49 

12 

.70  159 

.93  665 

.76  493 

.23  507 

.06  335 

.29841 

48 

13 

.70  180 

.93  658 

.76  522 

.23  478 

.06  342 

.29  820 

45 

14 

.70  202 

.93  650 

.76  551 

.23  449 

.06  350 

.29  798 

46 

15 

9.70  224 

9.93  643 

9.76  580 

0.23  420 

0.06  357 

0.29  776 

45 

16 

.70  245 

.93  636 

.76  609 

.23  391 

.06  364 

.29  755 

44 

17 

.70  267 

.93  628 

.76  639 

.23  361 

.06  372 

.29  733 

43 

18 

.70  288 

.93  621 

.76  668 

.23  332 

.06  379 

.29  712 

42 

19 

.70  310 

.93  614 

.76  697 

.23  303 

.06  386 

.29  690 

41 

20 

9.70  332 

9.93  606 

9.76  725 

0.23  275 

0.06  394 

0.29  668 

40 

21 

.70  353 

.93  599 

.76  754 

.23  246 

.06  401 

.29  647 

39 

22 

.70  375 

.93  591 

.76  783 

.23  217 

.06  409 

.29  625 

38 

23 

.70  396 

.93  584 

.76  812 

.23  188 

.06  416 

.29  604 

37 

24 

.70  418 

.93  577 

.76  841 

.23  159 

.06  423 

.29  582 

36 

25 

9.70  439 

9.93  569 

9.76  870 

0.23  130 

0.06  431 

0.29  561 

35 

26 

.70  461 

.93  562 

.76  899 

.23  101 

.06  438 

.29  539 

34 

27 

.70  482 

.93  554 

.76  928 

.23  072 

.06  446 

.29  518 

33 

28 

.70  504 

.93  547 

.76  957 

.23  043 

.06  453 

.29  496 

32 

29 

.70  525 

.93  539 

.76  986 

.23  014 

.06  461 

.29  475 

31 

30 

9.70  547 

9.93  532 

9.77  015 

0.22  985 

0.06  468 

0.29  453 

30 

31 

.70  568 

.93  525 

.77  044 

.22  956 

.06  475 

.29  432 

29 

32 

.70  590 

.93  517 

.77  073 

.22  927 

.06  483 

.29  410 

28 

33 

.70611 

.93  510 

.77  101 

.22  899 

.06  490 

.29  389 

27 

34 

.70  633 

.93  502 

.77  130 

.22  870 

.06  498 

.29  367 

26 

35 

9.70  654 

9.93  495 

9.77  159 

0.22  841 

0.06  505 

0.29  346 

25 

36 

.70  675 

.93  487 

.77  188 

.22  812 

.06  513 

.29  325 

24 

37 

.70  697 

.93  480 

.77  217 

.22  783 

.06  520 

.29  303 

23 

38 

.70  718 

.93  472 

.77  246 

.22  754 

.06  528 

.29  282 

22 

39 

.70  739 

.93  465 

.77  274 

.22  726 

.06  535 

.29  261 

21 

40 

9.70  761 

9.93  457 

9.77  303 

0.22  697 

0.06  543 

0.29  239 

20 

41 

.70  782 

.93  450 

.77  332 

.22  668 

.06  550 

.29  218 

19 

42 

.70  803 

.93  442 

.77  361 

.22  639 

.06  558 

.29  197 

18 

43 

.70  824 

.93  435 

.77  390 

.22  610 

.06  565 

.29  176 

17- 

44 

.70  846 

.93  427 

.77  418 

.22  582 

.06  573 

.29  154 

16 

45 

9.70  867 

9.93  420 

9.77  447 

0.22  553 

0.06  580 

0.29  133 

15 

46 

.70  888 

.93  412 

.77  476 

.22  524 

.06  588 

.29  112 

14 

47 

.70  909 

.93  405 

.77  505 

.22  495 

.06  595 

.29  091 

13 

48 

.70  931 

.93  397 

.77  533 

.22  467 

.06  603 

.29  069 

12 

49 

.70  952 

.93  390 

.77  562 

.22  438 

.06  610 

.29  048 

11 

50 

9.70  973 

9.93  382 

9.77  591 

0.22  409 

0.06  618 

0.29  027 

10 

51 

.70  994 

.93  375 

.77  619 

.22  381 

.06  625 

.29  006 

9 

52 

.71  015 

.93  367 

.77  648 

.22  352 

.06  633 

.28  985 

8 

53 

.71  036 

.93  360 

.77  677 

.22  323 

.06  640 

.28  964 

7 

54 

.71  058 

.93  352 

.77  706 

.22  294 

.06  648 

.28  942 

6 

55 

9.71  079 

9.93  344 

9.77  734 

0.22  266 

0.06  656 

0.28  921 

5 

56 

.71  100 

.93  337 

.77  763 

.22  237 

.06  663 

.28  900 

4 

57 

.71  121 

.93  329 

.77  791 

.22  209 

.06  671 

.28  879 

3 

58 

.71  142 

.93  322 

.77  820 

.22  180 

.06  678 

.28  858 

2 

59 

.71  163 

.93  314 

.77  849 

.22  151 

.06  686 

.28837 

1 

60 

9.71  184 

9.93  307 

9.77  877 

0.22  123 

0.06  693 

0.28816 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

120°  (300°) 


(239°)  59° 


Table  4.    Trigonometric  Logarithms 


227 


31°  (211°) 


(328°)  148C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.71  184 

9.93  307 

9.77  877 

0.22  123 

0.06  693 

0.28  816 

60 

1 

.71  205 

.93  299 

.77  906 

.22  094 

.06  701 

.28  795 

59 

2 

.71  226 

.93  291 

.77  935 

.22  065 

.06  709 

.28  774 

58 

3 

.71  247 

.93284 

.77  963 

.22  037 

.06  716 

.28  753 

57 

4 

.71  268 

.93  276 

.77  992 

.22  008 

.06  724 

.28  732 

56 

5 

9.71  289 

9.93  269 

9.78  020 

0.21  980 

0.06  731 

0.28711 

55 

6 

.71  310 

.93  261 

.78  049 

.21  951 

.06  739 

.28  690 

54 

7 

.71  331 

.93  253 

.78  077 

.21  -923 

.06  747 

.28  669 

53 

8 

.71  352 

.93  246 

.78  106 

.21  894 

.06  754 

.28  648 

52 

9 

.71  373 

.93  238 

.78  135 

.21  865 

.06  762 

.28  627 

51 

10 

9.71  393 

9.93  230 

9.78  163 

0.21  837 

0.06  770 

0.28  607 

50 

11 

.71  414 

.93  223 

.78  192 

'.21  808 

.06  777 

.28  586 

49 

12 

.71  435 

.93  215 

.78  220 

.21  780 

.06  785 

.28  565 

48 

13 

.71  456 

.93  207 

.78  249 

.21  751 

.06  793 

.28  544 

47 

14 

.71  477 

.93  200 

.78  277 

.21  723 

.06  800 

.28  523 

46 

15 

9.71  498 

9.93  192 

9.78  306 

0.21  694 

0.06  808 

0.28  502 

45 

16 

.71  519 

.93184 

.78  334 

.21  666 

.06  816 

.28  481 

44 

17 

.71  539 

.93  177 

.78  363 

.21  637 

.06  823 

.28  461 

43 

18 

.71  560 

.93  169 

.78  391 

.21  609 

.06  831 

.28440 

42 

19 

.71  581 

.93  161 

.78  419 

.21  581 

.06  839 

.28  419 

41 

20 

9.71  602 

9.93  154 

.78  448 

0.21  552 

0.06  846 

0.28  398 

40 

21 

.71  622 

.93  146 

.78  476 

.21  524 

.06  854 

.28  378 

39 

22 

.71  643 

.93  138 

.78  505 

.21  495 

.06  862 

.28  357 

38 

23 

.71  664 

.93  131 

.78  533 

.21  467 

.06  869 

.28  336 

37 

24 

.71  685 

.93  123 

.78  562 

.21  438 

.06  877 

.28  315 

36 

25 

9.71  705 

9.93  115 

9.78  590 

0.21  410 

0.06  885 

0.28  295 

35 

26 

.71  726 

.93  108 

.78  618 

.21  382 

.06  892 

.28  274 

34 

27 

.71  747 

.93  100 

.78647 

.21  353 

.06  900 

.28  253 

33 

28 

.71  767 

.93  092 

.78  675 

.21  325 

.06  908 

.28  233 

32 

29 

.71  788 

.93084 

.78  704 

.21  296 

.06  916 

.28  212 

31 

30 

9.71  809 

9.93  077 

9.78  732 

0.21  268 

0.06  923 

0.28  191 

30 

31 

.71  829 

.93  069 

.78  760 

.21  240 

.06  931 

.28  171 

29 

32 

.71  850 

.93  061 

.78  789 

.21  211 

.06  939 

.28  150 

28 

33 

.71  870 

.93  053 

.78  817 

.21  183 

.06  947 

.28  130 

27 

34 

.71  891 

.93046 

.78845 

.21  155 

.06  954 

.28  109 

26 

35 

9.71  911 

9.93  038 

9.78  874 

0.21  126 

0.06  962 

0.28  089 

25 

36 

.71  932 

.93  030 

.78  902 

.21  098 

.06  970 

.28  068 

24 

37 

.71  952 

.93  022 

.78  930 

.21  070 

.06  978 

.28  048 

23 

38 

.71  973 

.93  014 

.78  959 

.21041 

.06  986 

.28  027 

22 

39 

.71  994 

.93  007 

.78  987 

.21  013 

.06  993 

.28  006 

21 

40 

9.72  014 

9.92  999 

9.79  015 

0.20  985 

0.07  001 

0.27  986 

20 

41 

.72  034 

.92  991 

.79  043 

.20  957 

.07  009 

.27  966 

19 

42 

.72  055 

.92  983 

.79  072 

.20  928 

.07  017 

.27  945 

18 

43 

.72  075 

.92  976 

.79  100 

.20900 

.07  024 

.27  925 

17 

44 

.72  096 

.92  968 

.79  128 

.20  872 

.07  032 

.27  904 

16 

45 

9.72  116 

9.92  960 

9.79  156 

0.20844 

0.07  040 

0.27  884 

15 

46 

.72  137 

.92  952 

.79  185 

.20  815 

.07  048 

.27  863 

14 

47 

.72  157 

.92  944 

.79  213 

.20  787 

.07  056 

.27843 

13 

48 

.72  177 

.92  936 

.79  241 

.20  759 

.07  064 

.27  823 

12 

49 

.72  198 

.92  929 

.79  269 

.20  731 

.07  071 

.27  802 

11 

50 

9.72  218 

9.92  921 

9.79  297 

0.20  703 

0.07  079 

0.27  782 

10 

51 

.72  238 

.92  913 

.79  326 

.20  674 

.07  087 

.27  762 

9 

52 

.72  259 

.92  905 

.79  354 

.20646 

.07  095 

.27  741 

8 

53 

.72  279 

.92  897 

.79  382 

.20  618 

.07  103 

.27  721 

7 

54 

.72  299 

.92  889 

.79  410 

.20  590 

.07  111 

.27  701 

6 

55 

9.72  320 

9.92  881 

9.79  438 

0.20  562 

0.07  119 

0.27  680 

5 

56 

.72  340 

.92  874 

.79  466 

.20  534 

.07  126 

.27  660 

4 

57 

.72  360 

.92  866 

.79  495 

.20  505 

.07  134 

.27  640 

3 

58 

.72  381 

.92  858 

.79  523 

.20  477 

.07  142 

.27  619 

2 

59 

.72  401 

.92  850 

.79  551 

.20  449 

.07  150 

.27  599 

1 

60 

9.72  421 

9.92  842 

9.79  579 

0.20  421 

0.07  158 

0.27  579 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

121°  (301°) 


(238°)  58° 


228 


Table  4.    Trigonometric  Logarithms 


32°  (212°) 


(327°)  147< 


/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.72  421 

9.92  842 

9.79  579 

0.20  421 

0.07  158 

0.27  579 

60 

1 

.72  441 

.92  834 

.79  607 

.20  393 

.07  166 

.27  559 

59 

2 

.72  461 

.92  826 

.79  635 

.20  365 

.07  174 

.27  539 

58 

3 

.72  482 

.92  818 

.79  663 

.20  337 

.07  182 

.27  518 

57 

4 

.72  502 

.92  810 

.79  691 

.20  309 

.07  190 

.27  498 

56 

5 

9.72  522 

9.92  803 

9.79  719 

0.20  281 

0.07  197 

0.27  478 

55 

6 

.72  542 

.92  795 

.79  747 

.20  253 

.07  205 

.27  458 

54 

7 

.72  562 

.92  787 

.79  776 

.20  224 

.07  213 

.27  438 

53 

8 

.72  582 

.92  779 

.79  804 

.20  196 

.07  221 

.27  418 

52 

9 

.72  602 

.92  771 

.79  832 

.20  168 

.07  229 

.27  398 

51 

10 

9.72  622 

9.92  763 

9.79  860 

0.20  140 

0.07  237 

0.27  378 

50 

11 

.72  643 

.92  755 

.79  888 

.20  112 

.07  245 

.27  357 

49 

12 

.72  663 

.92  747 

.79  916 

.20  084 

.07  253 

.27  337 

48 

13 

.72  683 

.92  739 

.79  944 

.20  056 

.07  261 

.27  317 

47 

14 

.72  703 

.92  731 

.79  972 

.20  028 

.07  269 

.27  297 

46 

15 

9.72  723 

9.92  723 

9.80  000 

0.20  000 

0.07  277 

0.27  277 

45 

16 

.72  743 

.92  715 

.80  028 

.19  972 

.07285 

.27  257 

44 

17 

.72  763 

.92  707 

.80  056 

.19  944 

.07  293 

.27  237 

43 

18 

•72  783 

.92  699 

.80  084 

.19916 

.07  301 

.27  217 

42 

19 

.72803 

.92  691 

.80  112 

.19  888 

.07  309 

.27  197 

41 

20 

9.72  823 

9.92  683 

9.80  140 

0.19  860 

0.07  317 

0.27  177 

40 

21 

.72  843 

.92  675 

.80  168 

.19  832 

.07  325 

.27  157 

39 

22 

.72  863 

.92  667 

.80  195 

.19  805 

.07  333 

.27  137 

38 

23 

.72  883 

.92  659 

.80  223 

.19  777 

.07  341 

.27  117 

37 

24 

.72  902 

.92  651 

.80  251 

.19  749 

.07  349 

.27  098 

36 

25 

9.72  922 

9.92  643 

9.80  279 

0.19  721 

0.07  357 

0.27  078 

35 

26 

.72  942 

.92  635 

.80  307 

.19  693 

.07  365 

.27  058 

34 

27 

.72  962 

.92  627 

.80  335 

.19  665 

.07  373 

.27  038 

33 

28 

.72  982 

.92  619 

.80  363 

.19  637 

.07  381 

.27  018 

32 

29 

.73  002 

.92611 

.80  391 

.19  609 

.07  389 

.26  998 

31 

30 

9.73  022 

0.92  603 

9.80  419 

0.19  581 

0.07  397 

0.26  978 

30 

31 

.73  041 

.92  595 

.80  447 

.19  553 

.07  405 

.26  959 

29 

32 

•73  061 

.92  587 

.80  474 

.19  526 

.07413 

.26  939 

28 

33 

.73  081 

.92  579 

.80  502 

.19  498 

.07  421 

.26  919 

27 

34 

.73  101 

.92  571 

.80  530 

.19  470 

.07  429 

.26  899 

26 

35 

9.73  121 

9.92  563 

9.80  558 

0.19  442 

0.07  437 

0.26  879 

25 

36 

.73  140 

.92  555 

.80  586 

.19414 

.07  445 

.26  860 

24 

37 

.73  160 

.92  546 

.80  614 

.19  386 

.07  454 

.26840 

23 

38 

.73  180 

.92  538 

.80  642 

.19  358 

.07  462 

.26  820 

22 

39 

.73  200 

.92  530 

.80  669 

.19  331 

.07  470 

.26  800 

21 

40 

9.73  219 

9.92  522 

9.80  697 

0.19  303 

0.07  478 

0.26  781 

20 

41 

.73  239 

.92  514 

.80  725 

.19  275 

.07  486 

.26  761 

19 

42 

.73  259 

.92  506 

.80  753 

.19  247 

.07  494 

.26  741 

18 

43 

.73  278 

.92  498 

.80  781 

.19219 

.07  502 

.26  722 

17 

44 

.73  298 

.92  490 

.80  808 

.19  192 

.07  510 

.26  702 

16 

45 

9.73  318 

9.92  482 

9.80  836 

0.19  164 

0.07  518 

0.26  682 

15 

46 

.73  337 

.92  473 

.80  864 

.19  136 

.07  527 

.26  663 

14 

47 

.73  357 

.92  465 

.80  892 

.19  108 

.07  535 

.26  643 

13 

48 

.73  377 

.92  457 

.80  919 

.19  081 

.07  543 

.26  623 

12 

49 

.73  396 

.92  449 

.80  947 

.19  053 

.07  551 

.26  604 

11 

50 

9.73  416 

9.92  441 

9.80  975 

0.19  025 

0.07  559 

0.26  584 

10 

51 

.73  435 

.92  433 

.81  003 

.18  997 

.07  567 

.26  565 

9 

52 

.73  455 

.92  425 

.81  030 

.18  970 

.07  575 

.26  545 

8 

53 

.73  474 

.92  416 

.81  058 

.18  942 

.07  584 

.26  526 

7 

54 

.73  494 

.92  408 

.81  086 

.18914 

.07  592 

.26  506 

6 

55 

9.73  513 

9.92  400 

9.81  113 

0.18  887 

0.07  600 

0.26  487 

5 

56 

.73  533 

.92  392 

.81  141 

.18  859 

.07  608 

.26  467 

4 

57 

.73  552 

.92  384 

.81  169 

.18831 

.07  616 

.26  448 

3 

58 

.73  572 

.92  376 

.81  196 

.18  804 

.07  624 

.26  428 

2 

59 

.73  591 

.92  367 

.81  224 

.18776 

.07  633 

.26  409 

1 

60 

9.73611 

9.92  359 

9.81  252 

0.18  748 

0.07  641 

0.26  389 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

122°  (302°) 


(237°)  57C 


Table  4.    Trigonometric  Logarithms 


229 


33°  (213°) 


(326°)  146C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.73611 

9.92  359 

9.81  252 

0.18748 

0.07  641 

0.26  389 

60 

1 

.73  630 

.92  351 

.81  279 

.18721 

.07  649 

.26  370 

59 

2 

.73  650 

.92  343 

.81  307 

.18  693 

.07  657 

.26  350 

58 

3 

.73  669 

.92  335 

.81  335 

.18  665 

.07  665 

.26  331 

57 

4 

.73  689 

.92  326 

.81  362 

.18  638 

.07  674 

.26311 

56 

5 

9.73  708 

9.92  318 

9.81  390 

0.18  610 

0.07  682 

0.26  292 

55 

6 

.73  727 

.92  310 

.81  418 

.18  582 

.07  690 

.26  273 

54 

7 

.73  747 

.92  302 

.81  445 

.18  555 

.07  698 

.26  253 

53 

8 

.73  766 

.92  293 

.81  473 

.18527 

.07  707 

.26  234 

52 

9 

.73  785 

.92  285 

.81  500 

.18  500 

.07  715 

.26  215 

51 

10 

9.73  805 

9.92  277 

9.81  528 

0.18  472 

0.07  723 

0.26  195" 

50 

11 

.73  824 

.92  269 

.81  556 

.18  444 

.07  731 

.26  176 

49 

12 

.73  843 

.92  260 

.81  583 

.18417 

.07  740 

.26  157 

48 

13 

.73  863 

.92  252 

.81  611 

.18  389 

.07  748 

.26  137 

47 

14 

.73  882 

.92  244 

.81  638 

.18362 

.07  756 

.26  118 

46 

15 

9.73  901 

9.92  235 

9.81  666 

0.18  334 

0.07  765 

0.26  099 

45 

16 

.73  921 

.92  227 

.81  693 

.18  307 

.07  773 

.26  079 

44 

17 

.73  940 

.92  219 

.81  721 

.18  279 

.07  781 

.26  060 

43 

18 

.73  959 

.92211 

.81  748 

.18252 

.07  789 

.26  041 

42 

19 

.73  978 

.92  202 

.81  776 

.18  224 

.07  798 

.26  022 

41 

20 

9.73  997 

9.92  194 

9.81  803 

0.18  197 

0.07  806 

0.26  003 

40 

21 

.74  017 

.92  186 

.81  831 

.18  169 

.07  814 

.25  983 

39 

22 

.74  036 

.92  177 

.81  858 

.18  142 

.07  823 

.25  964 

38 

23 

.74  055 

.92  169 

.81  886 

.18  114 

.07  831 

.25  945 

37 

24 

.74  074 

.92  161 

.81  913 

.18  087 

.07  839 

.25  926 

36 

25 

9.74  093 

9.92  152 

9.81  941 

0.18  059 

0.07  848 

0.25  907 

35 

26 

.74  113 

.92  144 

.81  968 

.18032 

.07  856 

.25  887 

34 

27 

.74  132 

.92  136 

.81  996 

.18  004 

.07  864 

.25  868 

33 

28 

.74  151 

.92  127 

.82  023 

.17  977 

.07  873 

.25849 

.  32 

29 

,74  170 

.92  119 

.82  051 

.17  949 

.07  881 

.25  830 

31 

30 

9.74  189 

9.92  111 

9.82  078 

0.17922 

0.07  889 

0.25  811 

30 

31 

.74  208 

.92  102 

.82  106 

.17  894 

.07  898 

.25  792 

29 

32 

.74  227 

.92  094 

.82  133 

.17  867 

.07  906 

.25  773 

28 

33 

.74  246 

.92  086 

.82  161 

.17  839 

.07  914 

.25  754 

27 

34 

.74  265 

.92  077 

.82  188 

.17812 

.07  923 

.25  735 

26 

35 

9.74  284' 

9.92  069 

9.82  215 

0.17  785 

0.07  931 

0.25  716 

25 

36 

.74  303 

.92  060 

.82  243 

.17  757 

.07  940 

.25  697 

24 

37 

.74  322 

.92  052 

.82  270 

.17  730 

.07  948 

.25  678 

23 

38 

.74  341 

.92  044 

.82  298 

.17  702 

.07  956 

,25  659  ' 

22 

39 

.74  360 

.92  035 

.82  325 

.17  675 

.07  965 

.25  640 

21 

40 

9.74  379 

9.92  027 

9.82  352 

0.17  648 

0.07  973 

0.25  621 

20 

41 

.74  398 

.92  018 

.82  380 

.17  620 

.07  982 

.25  602 

19 

42 

.74  417 

.92  010 

.82  407 

.17  593 

.07  990 

.25  583 

18 

43 

.74  436 

.92  002 

.82  435 

.17  565 

.07  998 

.25  564 

17 

44 

.74  455 

.91  993 

.82  462 

.17  538 

.08  007 

.25  545 

16 

45 

9.74  474 

9.91  985 

9.82  489 

0.17511 

0.08  015 

0.25  526 

15 

46 

.74  493 

.91  976 

.82  517 

.17483 

.08  024 

.25  507 

14 

47 

.74  512 

.91  968 

.82  544 

.17  456 

.08  032 

.25  488 

13 

48 

.74  531 

.91  959 

.82  571 

.17  429 

.08  041 

.25  469 

12 

49 

.74  549 

.91  951 

.82  599 

.17401 

.08  049 

.25  451 

11 

50 

9.74  568 

9.91  942 

9.82  626 

0.17  374 

0.08  058 

0.25  432 

10 

51 

.74  587 

.91  934 

.82  653 

.17  347 

.08  066 

.25  413 

9 

52 

.74  606 

.91  925 

.82  681 

.17319 

.08  075 

.25  394 

8 

53 

.74  625 

.91  917 

.82  708 

.17  292 

.08  083 

.25  375 

7 

54 

.74644 

.91  908 

.82  735 

.17  265 

.08  092 

.25  356 

6 

55 

9.74  662 

9.91  900 

9.82  762 

0.17  238 

0.08  100 

0.25  338 

5 

56 

.74  681 

.91  891 

.82  790 

.17210 

.08  109 

.25  319 

4 

57 

.74  700 

.91  883 

.82  817 

.17  183 

.08  117 

.25  300 

3 

58 

.74  719 

.91  874 

.82844 

.17  156 

.08  126 

.25  281 

2 

59 

.74  737 

.91  866 

.82  871 

.17  129 

.08  134 

.25  263 

1 

60 

9.74  756 

9.91  857 

9.82  899 

0.17  101 

0.08  143 

0.25  244 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

123°  (303°) 


(236°)  56C 


230 


Table  4.    Trigonometric  Logarithms 


34°  (214°) 


(325°)  145° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.74  756 

9.91  857 

9.82  899 

0.17  101 

0.08  143 

0.25  244 

60 

1 

.74  775 

.91  849 

.82  926 

.17  074 

.08  151 

.25  225 

59 

2 

.74  794 

.91  840 

.82  953 

.17  047 

.08  160 

.25  206 

58 

3 

.74  812 

.91  832 

.82  980 

.17020 

.08  168 

.25  188 

57 

4 

.74  831 

.91  823 

.83  008 

.16  992 

.08  177 

.25  169 

56 

5 

9.74  850 

9.91  815 

9.83  035 

0.16  965 

0.08  185 

0.25  150 

55 

6 

.74  868 

.91  806 

.83  062 

.16  938 

.08  194 

.25  132 

54 

7 

.74  887 

.91  798 

.83  089 

.16911 

.08  202 

.25  113 

53 

8 

.74  906 

.91  789 

.83  117 

.16  883 

.08211 

.25  094 

52 

9 

.74  924 

.91  781 

.83  144 

.16  856 

.08  219 

.25  076 

51 

10 

9.74  943 

9.91  772 

9.83  171 

0.16  829 

0.08  228 

0.25  057 

50 

11 

.74  961 

.91  763 

.83  198 

.16  802 

.08  237 

.25  039 

49 

12 

.74  980 

.91  755 

.83  225 

.16  775 

.08  245 

.25  020 

48 

13 

.74  999 

.91  746 

.83  252 

.16  748 

.08  254 

.25  001 

47 

14 

.75  017 

.91  738 

.83  280 

•16  720 

.08  262 

.24  983 

46 

15 

9.75  036 

9.91  729 

9.83  307 

0.16  693 

0.08  271 

0.24  964 

45 

16 

.75  054 

.91  720 

.83  334 

.16  666 

.08  280 

.24  946 

44 

17 

.75  073 

.91  712 

.83  361 

.16  639 

.08  288 

.24  927 

43 

18 

.75  091 

.91  703 

.83  388 

.16612 

.08  297 

.24  909 

42 

19 

.75  110 

.91  695 

.83  415 

.16  585 

.08  305 

.24  890 

41 

20 

9.75  128 

9.91  686 

9.83  442 

0.16  558 

0.08  314 

0.24  872 

40 

21 

.75  147 

.91  677 

.83  470 

.16  530 

.08  323 

.24  853 

39 

22 

.75  165 

.91  669 

.83  497 

.16  503 

.08  331 

.24  835 

38 

23 

.75  184 

.91  660 

.83  524 

.16476 

.08  340 

.24  816 

37 

24 

.75  202 

.91  651 

.83  551 

.16449 

.08  349 

.24  798 

36 

25 

9.75  221 

9.91  643 

9.83  578 

0.16422 

0.08  357 

0.24  779 

35 

26 

.75  239 

.91  634 

.83  605 

.16  395 

.08  366 

.24  761 

34 

27 

.75  258 

.91  625 

.83  632 

.16  368 

.08  375 

.24  742 

33 

28 

.75  276 

.91  617 

.83  659 

.16341 

.08  383 

.24  724 

32 

29 

.75  294 

.91  608 

.83  686 

.16314 

.08  392 

.24  706 

31 

30 

9.75  313 

9.91  599 

9.83  713 

0.16  287 

0.08  401 

0.24  687 

30 

31 

.75  331 

.91  591 

.83  740 

.16  260 

.08  409 

.24  669 

29 

32 

.75  350 

.91  582 

.83  768 

.16  232 

.08  418 

.24  650 

28 

33 

.75  368 

.91  573 

.83  795 

.16  205 

.08  427 

.24  632 

27 

34 

.75  386 

.91  565 

.83  822 

.16  178 

.08  435 

.24  614 

26 

35 

9.75  405 

9.91  556 

9.83  849 

0.16  151 

0.08  444 

0.24  595 

25 

36 

.75  423 

.91  547 

.83  876 

.16  124 

.08  453 

.24  577 

24 

37 

.75  441 

.91  538 

.83  903 

.16  097 

.08  462 

.24  559 

23 

38 

.75  459 

.91  530 

.83  930 

.16  070 

.08  470 

.24  541 

22 

39 

.75  478 

.91  521 

.83  957 

.16  043 

.08  479 

.24  522 

21 

40 

9.75  496 

9.91  512 

9.83  984 

0.16016 

Q.08  488 

0.24  504 

20 

41 

.75  514 

.91  504 

.84011 

.15  989 

.08  496 

.24  486 

19 

42 

.75  533 

.91  495 

.84  038 

.15  962 

.08  505 

.24  467 

18 

43 

.75  551 

.91  486 

.84  065 

.15  935 

.08  514 

.24  449 

17 

44 

.75  569 

.91  477 

.84  092 

.15  908 

.08  523 

.24  431 

16 

45 

9.75  587 

9.91  469 

9.84  119 

0.15  881 

0.08  531 

0.24  413 

15 

46 

.75  605 

.91  460 

.84  146 

.15  854 

.08  540 

.24  395 

14 

47 

.75  624 

.91  451 

.84  173 

.15  827 

.08  549 

.24  376 

13 

48 

.75  642 

.91  442 

.84200 

.15  800 

.08  558 

.24  358 

12 

49 

.75  660 

.91  433 

.84227 

.15  773 

.08  567 

.24  340 

11 

50 

9.75  678 

9.91  425 

9.84  254 

0.15  746 

0.08  575 

0.24  322 

10 

51 

.75  696 

.91  416 

.84  280 

.15  720 

.08  584 

.24  304 

9 

52 

.75  714 

.91  407 

.84307 

.15  693 

.08  593 

.24  286 

8 

53 

.75  733 

.91  398 

.84334 

.15  666 

.08  602 

.24  267 

7 

54 

.75  751 

.91  389 

.84361 

.15  639 

.08611 

.24  249 

6 

55 

9.75  769 

9.91  381 

9.84  388 

0.15  612 

0.08  619 

0.24  231 

5 

56 

.75  787 

.91  372 

.84415 

.15  585 

.08  628 

.24  213 

4 

57 

.75  805 

.91  363 

.84442 

.15  558 

.08  637 

.24  195 

3 

58 

.75  823 

.91  354 

.84  469 

.15531 

.08  646 

.24  177 

2 

59 

.75  841 

.91  345 

.84  496 

.15  504 

.08  655 

.24  159 

1 

60 

9.75  859 

9.91  336 

9.84  523 

0.15  477 

0.08  664 

0.24  141 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

124°  (304°) 


(235°)  55° 


Table  4.    Trigonometric  Logarithms 


231 


35°  (215°) 


(324°)  144° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.75  859 

9.91  336 

9.84  523 

0.15477 

0.08  664 

0.24  141 

60 

1 

.75  877 

.91  328 

.84550 

.15  450 

.08  672 

.24  123 

59 

2 

.75  895 

.91  319 

.84576 

.15  424 

.08  681 

.24  105 

58 

3 

.75  913 

.91  310 

.84603 

.15  397 

.08  690 

.24  087 

57 

4 

.75  931 

.91  301 

.84630 

.15  370 

.08  699 

.24  069 

56 

5 

9.75  949 

9.91  292 

9.84  657 

0.15  343 

0.08  708 

0.24  051 

55 

6 

.75  967 

.91  283 

.84684 

.15316 

.08  717 

.24  033 

54 

7 

.75  985 

.91  274 

.84711 

.15  289 

.08  726 

.24  015 

53 

8 

.76  003 

.91  266 

.84738 

.15  262 

.08  734 

.23  997 

52 

9 

.76  021 

.91  257 

.84764 

.15  236 

.08  743 

.23  979 

51 

10 

9.76  039 

9.91  248 

9.84  791 

0.15  209 

0.08  752 

0.23  961 

50 

11 

.76  057 

.91  239 

.84818 

.15  182 

.08  761 

.23  943 

49 

12 

.76  075 

.91  230 

.84845 

.15  155 

.08  770 

.23  925 

48 

13 

.76  093 

.91  221 

.84872 

.15  128 

.08  779 

.23  907 

47 

14 

.76  111 

.91  212 

.84899 

.15  101 

.08  788 

.23  889 

46 

15 

9.76  129 

9.91  203 

9.84  925 

0.15  075 

0.08  797 

0.23  871 

45 

16 

.76  146 

.91  194 

.84952 

.15  048 

.08  806 

.23  854 

44 

17 

.76164 

.91  185 

.84979 

.15  021 

.08  815 

.23  836 

43 

18 

.76  182 

.91  176 

.85  006 

.14  994 

.08  824 

.23  818 

42 

19 

.76  200 

.91  167 

.85  033 

.14  967 

.08  833 

.23  800 

41 

20 

9.76  218 

9.91  158 

9.85  059 

0.14  941 

0.08  842 

0.23  782 

40 

21 

.76  236 

.91  149 

.85  086 

.14914 

.08  851 

.23764 

39 

22 

.76  253 

.91  141 

.85  113 

.14  887 

.08  859 

.23  747 

38 

23 

.76  271 

.91  132 

.85  140 

.14  860 

.08  868 

.23  729 

37 

24 

.76  289 

.91  123 

.85  166 

.14  834 

.08  877 

.23711 

36 

25 

9.76  307 

9.91  114 

9.85  193 

0.14  807 

0.08  886 

0.23  693 

35 

26 

.76  324 

.91  105 

.85  220 

.14  780 

.08  895 

.23  676 

34 

27 

.76  342 

.91  096 

.85  247 

.14  753 

.08  904 

.23  658 

33 

28 

.76  360 

.91  087 

.85  273 

.14  727 

.08  913 

.23  640 

32 

29 

.76  378 

.91  078 

.85  300 

.14  700 

.08  922 

.23  622 

31 

30 

9.76  395 

9.91  069 

9.85  327 

0.14  673 

0.08  931 

0.23  605 

30 

31 

.76  413 

.91  060 

.85  354 

.14  646 

.08  940 

.23  587 

29 

32 

.76  431 

.91  051 

.85  380 

.14  620 

.08  949 

.23  569 

28 

33 

.76  448 

.91  042 

.85  407 

.14  593 

.08  958 

.23  552 

27 

34 

.76  466 

.91  033 

.85  434 

.14  566 

.08  967 

.23  534 

26 

35 

9.76  484 

9.91  023 

9.85  460 

0.14  540 

0.08  977 

0.23  516 

25 

36 

.76  501 

.91  014 

.85  487 

.14513 

.08  986 

.23  499 

24 

37 

.76  519 

.91  005 

.85  514 

.14  486 

.08  995 

.23  481 

23 

38 

.76  537 

.90  996 

.85  540 

.14  460 

.09  004 

.23  463 

22 

39 

.76  554 

.90  987 

.85  567 

.14433 

.09  013 

.23  446 

21 

40 

9.76  572 

9.90  978 

9.85  594 

0.14406 

0.09  022 

0.23  428 

20 

41 

.76  590 

.90  969 

.85  620 

.14  380 

.09  031 

.23  410 

19 

42 

.76  607 

.90  960 

.85  647 

.14  353 

.09  040 

.23  393 

18 

43 

.76  625 

.90  951 

.85674 

.14  326 

.09  049 

.23  375 

17 

44 

.76  642 

.90  942 

.85  700 

.14  300 

.09  058 

.23  358 

16 

45 

9.76  660 

9.90  933 

9.85  727 

0.14  273 

0.09  067 

0.23  340 

15 

46 

.76  677 

.90  924 

.85  754 

.14  246 

.09  076 

.23  323 

14 

47 

.76  695 

.90  915 

.85  780 

.14  220 

.09  085 

.23  305 

13 

48 

.76  712 

.90  906 

.85  807 

.14  193 

.09  094 

.23  288 

12 

49 

.76  730 

.90  896 

.85  834 

.14  166 

.09  104 

.23  270 

11 

50 

9.76  747 

9.90  887 

9.85  860 

0.14  140 

0.09  113 

0.23  253 

10 

51 

.76  765 

.90878 

.85  887 

.14  113 

.09  122 

.23  235 

9 

52 

.76  782 

.90  869 

.85913 

.14  087 

.09  131 

.23  218 

8 

53 

.76  800 

.90  860 

.85940 

.14  060 

.09  140 

.23  200 

7 

54 

.76  817 

.90  851 

.85  967 

.14  033 

.09  149 

.23  183 

6 

55 

9.76  835 

9.90  842 

9.85  993 

0.14  007 

0.09  158 

0.23  165 

5 

56 

.76  852 

.90  832 

.86  020 

.13  980 

.09  168 

.23  148 

4 

57 

.76  870 

.90  823 

.86  046 

.13  954 

.09  177 

.23  130 

3 

58 

.76  887 

.90  814 

.86  073 

.13  927 

.09  186 

.23  113 

2 

59 

.76  904 

.90  805 

.86  100 

.13  900 

.09  195 

.23  096 

1 

60 

9.76  922 

9.90  796 

9.86  126 

0.13  874 

0.09  204 

0.23  078 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

125°  (305°) 


(234°)  54° 


232 


Table  4.    Trigonometric  Logarithms 


36°  (216°) 


(323°)  143° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.76  922 

9.90  796 

9.86  126 

0.13  874 

0.09  204 

0.23  078 

60 

1 

.76  939 

.90  787 

.86  153 

.13  847 

.09  213 

.23  061 

59 

2 

.76  957 

.90  777 

.86  179 

.13821 

.09  223 

.23  043 

58 

3 

.76  974 

.90  768 

.86  206 

.13  794 

.09  232 

.23  026 

57 

4 

.76  991 

.90  759 

.86  232 

.13  768 

.09  241 

.23  009 

56 

5 

9.77  009 

9.90  750 

9.86  259 

0.13  741 

0.09  250 

0.22  991 

55 

6 

.77  026 

.90  741 

.86  285 

.13715 

.09  259 

.22  974 

54 

7 

.77  043 

.90  731 

.86  312 

.13  688 

.09  269 

.22  957 

53 

8 

.77  061 

.90  722 

.86  338 

.13  662 

.09  278 

.22  939 

52 

9 

.77  078 

.90  713 

.86  365 

.13  635 

.09  287 

.22  922 

51 

10 

9.77  095 

9.90  704 

9.86  392 

0.13  608 

0.09  296 

0.22  905 

50 

11 

.77  112 

.90  694 

.86  418 

.13  582 

.09  306 

.22  888 

49 

12 

.77  130 

.90  685 

.86  445 

.13  555 

.09  315 

.22  870 

48 

13 

.77  147 

.90  676 

.86  471 

.13  529 

.09  324 

.22  853 

47 

14 

.77  164 

.90  667 

.86  498 

.13  502 

.09  333 

.22  836 

46 

15 

9.77  181 

9.90  657 

9.86  524 

0.13  476 

0.09  343 

0.22  819 

45 

16 

.77  199 

.90  648 

.86  551 

.13  449 

.09  352 

.22  801 

44 

17 

.77  216 

.90  639 

.86  577 

.13  423 

.09  361 

.22  784 

43 

18 

.77  233 

.90  630 

.86  603 

.13  397 

.09  370 

.22  767 

42 

19 

.77  250 

.90  620 

.86  630 

.13  370 

.09  380 

.22  750 

41 

20 

9.77  268 

9.90611 

9.86  656 

0.13  344 

0.09  389 

0.22  732 

40 

21 

.77  285 

.90  602 

.86  683 

.13317 

.09  398 

.22  715 

39 

22 

.77  302 

.90  592 

.86  709 

.13  291 

.09  408 

.22  698 

38 

23 

.77  319 

.90  583 

.86  736 

.13  264 

.09  417 

.22  681 

37 

24 

.77  336 

.90  574 

.86  762 

.13238 

.09  426 

.22  664 

36 

25 

9.77  353 

9.90  565 

9.86  789 

0.13211 

0.09  435 

0.22  647 

35 

26 

.77  370 

.90  555 

.86  815 

.13  185 

.09  445 

.22  630 

34 

27 

.77  387 

.90  546 

.86842 

.13  158 

.09  454 

.22  613 

33 

28 

.77  405 

.90  537 

.86  868 

.13  132 

.09  463 

.22  595 

32 

29 

.77  422 

.90  527 

.86  894 

•13  106 

.09  473 

.22  578 

31 

30 

9.77  439 

9.90  518 

9.86  921 

0.13  079 

0.09  482 

0.22  561 

30 

31 

.77  456 

.90  509 

.86  947 

.13  053 

.09  491 

.22  544 

29 

32 

.77  473 

.90  499 

.86  974 

.13  026 

.09  501 

.22  527 

28 

33 

.77  490 

.90  490 

.87  000 

.13  000 

.09  510 

.22  510 

27 

34 

.77  507 

.90  480 

.87  027 

.12  973 

.09  520 

.22  493 

26 

35 

9.77  524 

9.90  471 

9.87  053 

0.12  947 

0.09  529 

0.22  476 

25 

36 

.77  541 

.90  462 

.87  079 

.12921 

.09  538 

.22  459 

24 

37 

.77  558 

.90  452 

.87  106 

.12  894 

.09  548 

.22  442 

23 

38 

.77  575 

.90  443 

.87  132 

.12  868 

.09  557 

.22  425 

22 

39 

.77  592 

.90  434 

.87  158 

.12  842 

.09  566 

.22  408 

21 

40 

9.77  609 

9.90  424 

9.87  185 

0.12815 

0.09  576 

0.22  391 

20 

41 

.77  626 

.90415 

.87211 

.12  789 

.09  585 

.22  374 

19 

42 

.77  643 

.90  405 

.87  238 

.12  762 

.09  595 

.22  357 

18 

43 

.77  660 

.90  396 

.87  264 

.12  736 

.09  604 

.22  340 

17 

44 

.77  677 

.90386 

.87  290 

.12710 

.09  614 

.22  323 

16 

45 

9.77  694 

9.90  377 

9.87317 

0.12  683 

0.09  623 

0.22  306 

15 

46 

.77711 

.90  368 

.87  343 

.12  657 

.09  632 

.22  289 

14 

47 

.77-728 

.90  358 

.87  369 

.12  631 

.09  642 

.22  272 

13 

48 

.77744 

.90  349 

.87  396 

.12  604 

.09  651 

.22  256 

12 

49 

.77.761 

.90  339 

.87  422  • 

.12  578 

.09  661 

.22  239 

11 

50 

9.77  778 

9.90  330 

9.87  448 

0.12  552 

0.09  670 

0.22  222 

10 

51 

.77795 

.90  320 

.87  475 

.12  525 

.09  680 

.22  205 

9 

52 

.77.812 

.90311 

.87  501 

.12  499 

.09  689 

.22  188 

8 

53 

.77  829 

.90  301 

.87  527 

.12  473 

.09  699 

.22  171 

7 

54 

.77846 

.90  292 

.87  554 

.12  446 

.09  708 

.22  154 

6 

55 

9.77  862 

9.90  282 

9.87  580 

0.12  420 

0.09  718 

0.22  138 

5 

56 

.77879 

.90  273 

.87  606 

.12  394 

.09  727 

.22  121 

4 

57 

.77  896 

.90  263 

.87  633 

.12  367 

.09  737 

.22  104 

3 

58 

.77  913 

.90  254 

.87  659 

.12  341 

.09  746 

.22  087 

2 

59 

.77  930 

.90  244 

.87  685 

.12315 

.09  756 

.22  070 

1 

60 

9.77  946 

9.90  235 

9.87711 

0.12  289 

0.09  765 

0.22  054 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

126°  (306°) 


(233°)  53° 


Table  4.    Trigonometric  Logarithms 


233 


37°  (217°) 


(322°)  142C 


' 

Sin 

Cos 

Tan 

Cot 

Sec      Csc 

0 

9.77  946 

9.90  235 

9.87711 

0.12  289 

0.09  765 

0.22  054 

60 

1 

.77  963 

.90  225 

.87  738 

.12  262 

.09  775 

.22  037 

59 

2 

.77  980 

.90  216 

.87  764 

.12236 

.09  784 

.22  020 

58 

3 

.77  997 

.90  206 

.87  790 

.12210 

.09  794 

.22  003 

57 

4 

.78  013 

.90  197 

.87  817 

.12  183 

.09  803 

.21  987 

56 

5 

9.78  030 

9.90  187 

9.87  843 

0.12  157 

0.09  813 

0.21  970 

55 

6 

.78  047 

.90  178 

.87  869 

.12  131 

.09  822 

.21  953 

54 

7 

.78  063 

.90  168 

.87  895 

.12  105 

.09  832 

.21  937 

53 

8 

.78  080 

.90  159 

.87  922 

.12  078 

.09  841 

.21  920 

52 

9 

.78  097 

.90  149 

.87  948 

.12  052 

.09  851 

.21  903 

51 

10 

9.78113 

9.90  139 

9.87  974 

0.12  026 

0.09  861 

0.21  887 

50 

11 

.78  130 

.90  130 

.88  000 

.12  000 

.09  870 

.21  870 

49 

12 

.78  147 

.90  120 

.88  027 

.11  973 

.09  880 

.21  853 

48 

13 

.78  163 

.90  111 

.88  053 

.11  947 

.09  889 

.21  837 

47 

14 

.78  180 

.90  101 

.88  079 

.11  921 

.09  899 

.21  820 

46 

15 

9.78  197 

9.90  091 

9.88  105 

0.11  895 

0.09  909 

0.21  803 

45 

16 

.78  213 

.90  082 

.88  131 

.11  869 

.09  918 

.21  787 

44 

17 

.78  230 

.90  072 

.88  158 

.11  842 

.09  928 

.21  770 

43 

18 

.78  246 

.90  063 

.88184 

.11  816 

.09  937 

.21  754 

42 

19 

.78  263 

.90  053 

.88  210 

.11  790 

.09  947 

.21  737 

41 

20 

9.78  280 

9.90  043 

9.88  236 

0.11  764 

0.09  957 

0.21  720 

40 

21 

.78  296 

.90  034 

.88  262 

.11  738 

.09  966 

.21  704 

39 

22 

.78313 

.90  024 

.88  289 

.11  711 

.09  976 

.21  687 

38 

23 

.78  329 

.90  014 

.88  315 

.11  685 

.09  986 

.21  671 

37 

24 

.78  346 

.90  005 

.88  341 

.11  659 

.09  995 

.21  654 

36 

25 

9.78  362 

9.89  995 

9.88  367 

0.11  633 

0.10  005 

0.21  638 

35 

26 

.78  379 

.89  985 

.88  393 

.11  607 

.10015 

.21  621 

34 

27 

.78  395 

.89  976 

.88  420 

.11  580 

.10  024 

.21  605 

33 

28 

.78412 

.89  966 

.88  446 

.11  554 

.10034 

.21  588 

32 

29 

.78  428 

.89  956 

.88  472 

.11  528 

.10  044 

.21  572 

31 

30 

9.78  445 

9.89  947 

9.88  498 

0.11  502 

0.10  053 

0.21  555 

30 

31 

.78  461 

.89  937 

.88  524 

.11  476 

.10  063 

.21  539 

29 

32 

.78  478 

.89  927 

.88  550 

.11450 

.10  073 

.21  522 

28 

33 

.78  494 

.89  918 

.88  577 

.11  423 

.10  082 

.21  506 

27 

34 

.78  510 

.89  908 

.88  603 

.11397 

.10  092 

.21  490 

26 

35 

9.78  527 

9.89  898 

9.88  629 

0.11  371 

0.10  102 

0.21  473 

25 

36 

.78  543 

.89  888 

.88  655 

.11  345 

.10  112 

.21  457 

24 

37 

.78  560 

.89  879 

.88  681 

.11  319 

.10  121 

.21  440 

23 

38 

.78  576 

.89  869 

.88  707 

.11  293 

.10  131 

.21  424 

22 

39 

.78  592 

.89  859 

.88733 

.11  267 

.10  141 

.21  408 

21 

40 

9.78  609 

9.89  849 

9.88  759 

0.11  241 

0.10  151 

0.21  391 

20 

41 

.78  625 

.89840 

.88  786 

.11  214 

.10  160 

.21  375 

19 

42 

.78  642 

.89  830 

.88812 

.11  188 

.10  170 

.21  358 

18 

43 

.78  658 

.89  820 

.88  838 

.11  162 

.10  180 

.21  342 

17 

44 

.78  674 

.89  810 

.88864 

.11  136 

.10  190 

.21  326 

16 

45 

9.78  691 

9.89  801 

9.88  890 

0.11  110 

0.10  199 

0.21  309 

15 

46 

.78  707 

.89  791 

.88  916 

.11  084 

.10  209 

.21  293 

14 

47 

.78  723 

.89  781 

.88  942 

.11058 

.10219 

.21  277 

13 

48 

.78  739 

.89  771 

.88  968 

.11  032 

.10229 

.21  261 

12 

49 

.78  756 

.89  761 

.88  994 

.11  006 

.10  239 

.21  244 

11 

50 

9.78  772 

9.89  752 

9.89  020 

0.10980 

0.10248 

0.21  228 

10 

51 

.78  788 

.89  742 

.89  046 

.10954 

.10  258 

.21  212 

9 

52 

.78  805 

.89  732 

.89  073 

.10927 

.10  268 

.21  195 

8 

53 

.78  821 

.89  722 

.89  099 

.10901 

.10  278 

.21  179 

7 

54 

.78  837 

.89  712 

.89  125 

.10  875 

.10  288 

.21  163 

6 

55 

9.78  853 

9.89  702 

9.89  151 

0.10849 

0.10  298 

0.21  147 

5 

56 

.78  869 

.89  693 

.89  177 

.10  823 

.10  307 

.21  131 

4 

57 

.78  886 

.89  683 

.89  203 

.10  797 

.10317 

.21  114 

3 

58 

.78  902 

.89  673 

.89  229 

.10  771 

.10327 

.21  098 

2 

59 

.78918 

.89  663 

.89  255 

.10  745 

.10337 

.21  082 

1 

60 

9.78  934 

9.89  653 

9.89  281 

0.10719 

0.10  347 

0.21  066 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

127°  (307°) 


(232°)  52° 


234 


Table  4.    Trigonometric  Logarithms 


38°  (218°) 


(321°)  141C 


' 

Sin 

Cos 

Tan 

Cot 

Sec      Csc 

0 

9.78  934 

9.89  653 

9.89  281 

0.10719 

0.10347 

0.21  066 

60 

1 

.78  950 

.89  643 

.89  307 

.10  693 

.10357 

.21  050 

59 

2 

.78  967 

.89  633 

.89333 

.10  667 

.10367 

.21  033 

58 

3 

.78  983 

.89  624 

.89  359 

.10  641 

.10376 

.21  017 

57 

4 

.78  999 

.89  614 

.89  385 

.10615 

.10  386 

.21  001 

56 

5 

9.79  015 

9.89  604 

9.89411 

0.10589 

0.10  396 

0.20  985 

55 

6 

.79  031 

.89  594 

.89  437 

.10  563 

.10  406 

.20  969 

54 

7 

.79  047 

.89  584 

.89  463 

.10  537 

.10416 

.20  953 

53 

8 

.79  063 

.89  574 

.89  489 

.10511 

.10426 

.20  937 

52 

9 

.79  079 

.89  564 

.89  515 

.10  485 

.10436 

.20  921 

51 

10 

9.79  095 

9.89  554 

9.89  541 

0.10459 

0.10446 

0.20  905 

50 

11 

.79  111 

.89  544 

.89  567 

.10  433 

.10  456 

.20  889 

49 

12 

.79  128 

.89  534 

.89  593 

.10  407 

.10  466 

.20  872 

48 

13 

.79  144 

.89  524 

.89  619 

.10381 

.10476 

.20  856 

47 

14 

.79  160 

.89  514 

.89  645 

.10  355 

.10  486 

.20840 

46 

15 

9.79  176 

9.89  504 

9.89  671 

0.10  329 

0.10  496 

0.20  824 

45 

16 

.79  192 

.89  495 

.89  697 

.10303 

.10505 

.20  808 

44 

17 

.79  208 

.89  485 

.89  723 

.10277 

.10515 

.20  792 

43 

18 

.79  224 

.89  475 

.89  749 

.10251 

.10  525 

.20  776 

42 

19 

.79  240 

.89  465 

.89  775 

.10  225 

.10535 

.20  760 

41 

20 

9.79  256 

9.89  455 

9.89  801 

0.10  199 

0.10  545 

0.20  744 

40 

21 

.79  272 

.89  445 

.89  827 

.10  173 

.10  555 

.20  728 

39 

22 

.79  288 

.89  435 

.89  853 

.10  147 

.10  565 

.20  712 

38 

23 

.79  304 

.89  425 

.89  879 

.10  121 

.10  575 

.20  696 

37 

24 

.79  319 

.89  415 

.89  905 

.10095 

.10  585 

.20  681 

36 

25 

9.79  335 

9.89  405 

9.89  931 

0.10  069 

0.10595 

0.20  665 

35 

26 

.79  351 

.89  395 

.89  957 

.10  043 

.10  605 

.20  649 

34 

27 

.79  367 

.89  385 

.89  983 

.10017 

.10615 

.20  633 

33 

28 

.79  383 

.89  375 

.90  009 

.09  991 

.10  625 

.20  617 

32 

29 

.79  399 

.89  364 

.90  035 

.09  965 

.10  636 

.20  601 

31 

30 

9.79415 

9.89  354 

9.90  061 

0.09  939 

0.10646 

0.20  585 

30 

31 

.79  431 

.89  344 

.90  086 

.09  914 

.10  656 

.20  569 

29 

32 

.79  447 

.89  334 

.90  112 

.09  888 

.10  666 

.20  553 

28 

33 

.79  463 

.89  324 

.90  138 

.09  862 

.10  676 

.20  537 

27 

34 

.79  478 

.89  314 

.90  164 

.09  836 

.10  686 

.20  522 

26 

35 

9.79  494 

9.89  304 

9.90  190 

0.09  810 

0.10  696 

0.20  506 

25 

36 

.79  510 

.89  294 

.90  216 

.09  784 

.10  706 

.20  490 

24 

37 

.79  526 

.89  284 

.90  242 

.09  758 

.10716 

.20  474 

23 

38 

.79  542 

.89  274 

.90  268 

.09  732 

.10  726 

.20  458 

22 

39 

.79  558 

.89  264 

.90  294 

.09  706 

.10  736 

.20  442 

21 

40 

9.79  573 

9.89  254 

9.90  320 

0.09  680 

O'lO  746 

0.20  427 

20 

41 

.79  589 

.89  244 

.90  346 

.09  654 

.10  756 

.20411 

19 

42 

.79  605 

.89  233 

.90  371 

.09  629 

.10  767 

.20  395 

18 

43 

.79  621 

.89  223 

.90  397 

.09  603 

.10  777 

.20  379 

17 

44 

.79  636 

.89  213 

.90  423 

.09  577 

.10  787 

.20  364 

16 

45 

9.79  652 

9.89  203 

9.90  449 

0.09  551 

0.10  797 

0.20  348 

15 

46 

.79  668 

.89  193 

.90  475 

.09  525 

.10  807 

.20  332 

14 

47 

.79  684 

.89  183 

.90  501 

.09  499 

.10817 

.20  316 

13 

48 

.79  699 

.89  173 

.90  527 

.09  473 

.10  827 

.20  301 

12 

49 

.79  715 

.89  162 

.90  553 

.09  447 

.10  838 

.20  285 

11 

50 

9.79  731 

9.89  152 

9.90  578 

0.09  422 

0.10  848 

0.20  269 

10 

51 

.79  746 

.89  142 

.90  604 

.09  396 

.10  858 

.20  254 

9 

52 

.79  762 

.89  132 

.90  630 

.09  370 

.10  868 

.20  238 

8 

53 

.79  778 

.89  122 

.90  656 

.09  344 

.10  878 

.20  222 

7 

54 

.79  793 

.89112 

.90  682 

.09  318 

.10  888 

.20  207 

6 

55 

9.79  809 

9.89  101 

9.90  708 

0.09  292 

0.10  899 

0.20  191 

5 

56 

.79  825 

.89  091 

.90  734 

.09  266 

.10  909 

.20  175 

4 

57 

.79840 

.89  081 

.90  759 

.09  241 

.10919 

.20  160 

3 

58 

.79  856 

.89  071 

.90  785 

.09  215 

.10  929 

.20  144 

2 

59 

.79  872 

.89  060 

.90811 

.09  189 

.10  940 

.20  128 

1 

60 

9.79  887 

9.89  050 

9.90  837 

0.09  163 

0.10  950 

0.20  113 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

128°  (308°) 


(231°)  51C 


Table  4.    Trigonometric  Logarithms 


235 


39°  (219°) 


(320°)  140C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.79  887 

9.89  050 

9.90  837 

0.09  163 

0.10950 

0.20  113 

60 

1 

.79  903 

.89  040 

.90  863 

.09  137 

.10  960 

.20  097 

59 

2 

.79  918 

.89  030 

.90  889 

.09  111 

.10  970 

.20  082 

58 

3 

.79  934 

.89  020 

.90  914 

.09  086 

.10  980 

.20  066 

57 

4 

.79  950 

.89  009 

.90  940 

.09060 

.10991 

.20  050 

56 

5 

9.79  965 

9.88  999 

9.90  966 

0.09  034 

0.11  001 

0.20  035 

55 

6 

.79  981 

.88  989 

.90  992 

.09  008 

.11  Oil 

.20  019 

54 

7 

.79  996 

.88  978 

.91  018 

.08  982 

.11  022 

.20004 

53 

8 

.80012 

.88  968 

.91  043 

.08  957 

.11  032 

.19  988 

52 

9 

.80027 

.88  958 

.91  069 

.08  931 

.11  042 

.19  973 

51 

10 

9.80  043 

9.88  948 

9.91  095 

0.08  905 

0.11  052 

0.19  957 

50 

11 

.80  058 

.88  937 

.91  121 

.08  879 

.11  063 

.19  942 

49 

12 

.80  074 

.88  927 

.91  147 

.08  853 

.11  073 

.19  926 

48 

13 

.80  089 

.88  917 

.91  172 

.08  828 

.11  083 

.19911 

47 

14 

.80  105 

.88  906 

.91  198 

.08  802 

.11  094 

.19  895 

46 

15 

9.80  120 

9.88  896 

9.91  224 

0.08  776 

0.11  104 

0.19  880 

45 

16 

.80  136 

.88  886 

.91  250 

.08  750 

.11  114 

.19  864 

44 

17 

.80  151 

.88  875 

.91  276 

.08  724 

.11  125 

.19  849 

43 

18 

.80  166 

.88  865 

.91  301 

.08  699 

.11  135 

.19  834 

42 

19 

.80  182 

.88855 

.91  327 

.08  673 

.11  145 

.19818 

41 

20 

9.80  197 

9.88  844 

9.91  353 

0.08  647 

0.11  156 

0.19  803 

40 

21 

.80213 

.88  834 

.91  379 

.08  621 

.11  166 

.19  787 

39 

22 

.80  228 

.88  824 

.91  404 

.08  596 

.11  176 

.19  772 

38 

23 

.80  244 

.88  813 

.91  430 

.08  570 

.11  187 

.19  756 

37 

24 

.80  259 

.88  803 

.91456 

.08  544 

.11  197 

.19  741 

36 

25 

9.80  274 

9.88  793 

9.91  482 

0.08  518 

0.11  207 

0.19  726 

35 

26 

.80  290 

.88  782 

.91  507 

.08  493 

.11218 

.19710 

34 

27 

.80  305 

.88  772 

.91  533 

.08  467 

.11  228 

.19  695 

33 

28 

.80  320 

.88  761 

.91  559 

.08  441 

.11  239 

.19  680 

32 

29 

.80  336 

.88  751 

.91  585 

.08  415 

.11  249 

.19  664 

31 

30 

9.80  351 

9.88  741 

9.91  610 

0.08  390 

0.11  259 

0.19  649 

30 

31 

.80  366 

.88730 

.91  636 

.08364 

.11  270 

.19  634 

29 

32 

.80  382 

.88  720 

.91  662 

.08  338 

.11  280 

.19618 

28 

33 

.80  397 

.88  709 

.91  688 

.08  312 

.11  291 

.19  603 

27 

34 

.80  412 

.88  699 

.91  713 

.08  287 

.11  301 

.19  588 

26 

35 

9.80  428 

9.88  688 

9.91  739 

0.08  261 

0.11312 

0.19  572 

25 

36 

.80  443 

.88  678 

.91  765 

.08  235 

.11  322 

.19  557 

24 

37 

.80  458 

.88  668 

.91  791 

.08  209 

.11  332 

.19  542 

23 

38 

.80  473 

.88  657 

.91  816 

.08  184 

.11  343 

.19  527 

22 

39 

.80  489 

.88  647 

.91  842 

.08  158 

.11  353 

.19511 

21 

40 

9.80  504 

9.88  636 

9.91  868 

0.08  132 

0.11  364 

0.19496 

20 

41 

.80  519 

.88  626 

.91  893 

.08  107 

.11  374 

.19  481 

19 

42 

.80  534 

.88  615 

.91  919 

.08  081 

.11  385 

.19  466 

18 

43 

.80  550 

.88605 

.91  945 

.08  055 

.11  395 

.19  450 

17 

44 

.80  565 

.88  594 

.91  971 

.08  029 

.11  406 

.19435 

16 

45 

9.80  580 

9.88  584 

9.91  996 

0.08  004 

0.11416 

0.19  420 

15 

46 

.80  595 

.88  573 

.92  022 

.07  978 

.11  427 

.19  405 

14 

47 

.80  610 

.88  563 

.92  048 

.07  952 

.11437 

.19  390 

13 

48 

.80  625 

.88  552 

.92  073 

.07  927 

.11  448 

.19  375 

12 

49 

.80641 

.88542 

.92  099 

.07  901 

.11458 

.19  359 

11 

50 

9.80  656 

9.88  531 

9.92  125 

0.07  875 

0.11  469 

0.19  344 

10 

51 

.80  671 

.88  521 

.92  150 

.07  850 

.11  479 

.19329 

9 

52 

.80  686 

.88  510 

.92  176 

.07  824 

.11  490 

.19314 

8 

53 

.80  701 

.88  499 

.92  202 

.07  798 

.11  501 

.19  299 

7 

54 

.80716 

.88  489 

.92  227 

.07  773 

.11  511 

.19284 

6 

55 

9.80  731 

9.88  478 

9.92  253 

0.07  747 

0.11  522 

0.19  269 

5 

56 

.80746 

.88  468 

.92  279 

.07  721 

.11  532 

.19  254 

4 

57 

.80  762 

.88  457 

.92  304 

.07  696 

.11  543 

.19  238 

3 

58 

.80777 

.88  447 

.92  330 

.07  670 

.11  553 

.19  223 

2 

59 

.80  792 

.88  436 

.92  356 

.07644 

.11  564 

.19  208 

1 

60 

9.80  807 

9.88  425 

9.92  381 

0.07  619 

0.11  575 

0.19  193 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

129°  (309°) 


(230°)  50° 


236 


Table  4.    Trigonometric  Logarithms 


40°  (220°) 


(319°)  139C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.80  807 

9.88  425 

9.92  381 

0.07  619 

0.11  575 

0.19  193 

60 

1 

.80  822 

.88  415 

.92  407 

.07  593 

.11  585 

.19  178 

59 

2 

.80  837 

.88  404 

.92  433 

.07  567 

.11  596 

.19  163 

58 

3 

.80  852 

.88  394 

.92  458 

.07  542 

.11  606 

.19  148 

57 

4 

.80  867 

.88  383 

.92  484 

.07  516 

.11  617 

.19  133 

56 

5 

9.80  882 

9.88  372 

9.92  510 

0.07  490 

0.11  628 

0.19  118 

55 

6 

.80  897 

.88  362 

.92  535 

.07  465 

.11  638 

.19  103 

54 

7 

.80  912 

.88  351 

.92  561 

.07  439 

.11  649 

.19  088 

53 

8 

.80  927 

.88  340 

.92  587 

.07  413 

.11  660 

.19  073 

52 

9 

.80  942 

.88  330 

.92  612 

.07  388 

.11  670 

.19  058 

51 

10 

9.80  957 

9.88319 

9.92  638 

0.07  362 

0.11  681 

0.19043 

50 

11 

.80  972 

.88  308 

.92  663 

.07  337 

.11  692 

.19  028 

49 

12 

.80  987 

.88  298 

.92  689 

.07311 

.11  702 

.19013 

48 

13 

.81  002 

.88  287 

.92  715 

.07  285 

.11713 

.18  998 

47 

14 

.81  017 

.88  276 

.92  740 

•07  260 

.11  724 

.18983 

46 

15 

9.81  032 

9.88  266 

9.92  766 

0.07  234 

0.11  734 

0.18  968 

45 

16 

.81  047 

.88  255 

.92  792 

.07  208 

.11  745 

.18  953 

44 

17 

.81  061 

.88  244 

.92  817 

.07  183 

.11  756 

.18  939 

43 

18 

.81  076 

.88  234 

.92  843 

.07  157 

.11  766 

.18  924 

42 

19 

.81  091 

.88  223 

.92  868 

•07  132 

.11  777 

.18  909 

41 

20 

9.81  106 

9.88  212 

9.92  894 

0.07  106 

0.11  788 

0.18  894 

40 

21 

.81  121 

.88  201 

.92  920 

.07  080 

.11  799 

.18  879 

39 

22 

.81  136 

.88  191 

.92  945 

.07  055 

.11  809 

.18  864 

38 

23 

.81  151 

.88  180 

.92  971 

.07  029 

.11  820 

.18849 

37 

24 

.81  166 

.88  169 

.92  996 

.07  004 

.11  831 

.18  834 

36 

25 

9.81  180 

9.88  158 

9.93  022 

0.06  978 

0.11  842 

0.18  820 

35 

26 

.81  195 

.88  148 

.93  048 

.06  952 

.11  852 

.18  805 

34 

27 

.81  210 

.88  137 

.93  073 

.06  927 

.11  863 

.18  790 

33 

28 

.81  225 

.88  126 

.93  099 

.06  901 

.11  874 

.18  775 

32 

29 

.81  240 

.88  115 

.93  124 

.06  876 

.11  885 

.18  760 

31 

30 

9.81  254 

9.88  105 

9.93  150 

0.06  850 

0.11  895 

0.18  746 

30 

31 

.81  269 

.88  094 

.93  175 

.06  825 

.11  906 

.18731 

29 

32 

.81  284 

.88  083 

.93  201 

.06  799 

.11917 

.18716 

28 

33 

.81  299 

.88  072 

.93  227 

.06  773 

.11  928 

.18701 

27 

34 

.81  314 

.88  061 

.93  252 

.06  748 

.11  939 

.18686 

26 

35 

9.81  328 

9.88  051 

9.93  278 

0.06  722 

0.11  949 

0.18  672 

25 

36 

.81  343 

.88  040 

.93  303 

.06  697 

.11  960 

.18657 

24 

37 

.81  358 

.88  029 

.93  329 

.06  671 

.11  971 

.18642 

23 

38 

.81  372 

.88  018 

.93  354 

.06  646 

.11  982 

.18628 

22 

39 

.81  387 

.88  007 

.93  380 

.06  620 

.11  993 

.18613 

21 

40 

9.81  402 

9.87  996 

9.93  406 

0.06  594 

0.12  004 

0.18598 

20 

41 

.81  417 

.87  985 

.93  431 

.06  569 

.12015 

.18  583 

19 

42 

.81  431 

.87  975 

.93  457 

.06  543 

.12  025 

.18  569 

18 

43 

.81  446 

.87  964 

.92  482 

.06  518 

.12  036 

.18  554 

17 

44 

.81  461 

.87  953 

.93  508 

.06  492 

.12  047 

.18539 

16 

45 

9.81  475 

9.87  942 

9.93  533 

0.06  467 

0.12  058 

0.18  525 

15 

46 

.81  490 

.87  931 

.93  559 

.06  441 

.12  069 

.18510 

14 

47 

.81  505 

.87  920 

.93584 

.06  416 

.12  080 

.18495 

13 

48 

.81  519 

.87  909 

.93  610 

.06  390 

.12091 

.18481 

12 

49 

.81  534 

.87  898 

.93  636 

.06  364 

.12  102 

.18466 

11 

50 

9.81  549 

9.87  887 

9.93  661 

0.06  339 

0.12113 

0.18451 

10 

51 

.81  563 

.87  877 

.93  687 

.06  313 

.12  123 

.18437 

9 

52 

.81  578 

.87  866 

.93  712 

.06  288 

.12  134 

.18422 

8 

53 

.81  592 

.87  855 

.93  738 

.06  262 

.12  145 

.18  408 

7 

54 

.81  607 

.87844 

.93  763 

.06  237 

.12  156 

.18393 

6 

55 

9.81  622 

9.87  833 

9.93  789 

0.06211 

0.12  167 

0.18378 

5 

56 

.81  636 

.87  822 

.93  814 

.06  186 

.12  178 

.18  364 

4 

57 

.81  651 

.87811 

.93  840 

.06  160 

.12  189 

..18349 

3 

58 

.81  665 

.87  800 

.93  865 

.06  135 

.12  200 

.18335 

2 

59 

.81  680 

.87  789 

.93  891 

.06  109 

.12211 

.18320 

1 

60 

9.81  694 

9.87  778 

9.93  916 

0.06  084 

0.12  222 

.18  306 

0 

Cos 

Sin 

Cot   |   Tan 

Csc 

Sec 

' 

130°  (310°) 


(229°)  49 < 


Table  4.    Trigonometric  Logarithms 


237 


41°  (221°) 


(318°)  138° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.81  694 

9.87  778 

9.93  916 

0.06  084 

0.12222 

0.18  306 

60 

1 

.81  709 

.87  767 

.93  942 

.06  058 

.12  233 

.18291 

59 

2 

.81  723 

.87  756 

.93  967 

.06  033 

.12  244 

.18  277 

58 

3 

.81  738 

.87  745 

.93  993 

.06007 

.12255 

.18  262 

57 

4 

.81  752 

.87  734 

.94  018 

.05982 

.12  266 

.18  248 

56 

5 

9.81  767 

9.87  723 

9.94  044 

0.05  956 

0.12  277 

0.18233 

55 

6 

.81  781 

.87  712 

.94  069 

.05  931 

.12  288 

.18219 

54 

7 

.81  796 

.87  701 

.94095 

.05  905 

.12  299 

.18  204 

53 

8 

.81  810 

.87  690 

.94  120 

.05  880 

.12310 

.18  190 

52 

9 

.81  825 

.87  679 

.94  146 

.05  854 

.12321 

.18  175 

51 

10 

9.81  839 

9.87  668 

9.94  171 

0.05  829 

0.12  332 

0.18  161 

50 

11 

.81854 

.87  657 

.94  197 

.05  803 

.12  343 

.18  146 

49 

12 

.81  868 

.87  646 

.94  222 

.05  778 

.12  354 

.18  132 

48 

13 

.81  882 

.87  635 

.94  248 

.05  752 

.12  365 

.18118 

47 

14 

.81  897 

.87  624 

.94  273 

.05  727 

.12376 

.18  103 

46 

15 

9.81  911 

9.87  613 

9.94  299 

0.05  701 

0.12  387 

0.18  089 

45 

16 

.81  926 

.87  601 

.94324 

.05  676 

.12  399 

.18074 

44 

17 

.81  940 

.87  590 

.94  350 

.05  650 

.12410 

48  060 

43 

18 

.81  955 

.87  579 

.94  375 

.05  625 

.12421 

.18  045 

42 

19 

.81  969 

.87  568 

.94  401 

.05  599 

.12  432 

.18031 

41 

20 

9.81  983 

9.87  557 

9.94  426 

0.05  574 

0.12443 

0.18017 

40 

21 

.81  998 

.87  546 

.94  452 

.05  548 

.12  454 

.18  002 

39 

22 

.82  012 

.87  535 

.94477 

.05  523 

.12  465 

.17  988 

38 

23 

.82  026 

.87  524 

.94  503 

.05  497 

.12  476 

.17  974 

37 

24 

.82041 

.87  513 

.94  528 

.05  472 

.12  487 

.17  959 

36 

25 

9.82  055 

9.87  501 

9.94  554 

0.05  446 

0.12  499 

0.17  945 

35 

26 

.82  069 

.87  490 

.94579 

.05  421 

.12510 

.17931 

34 

27 

.82084 

.87  479 

.94604 

.05  396 

.12521 

.17916 

33 

28 

.82  098 

.87  468 

.94  630 

.05  370 

.12  532 

.17  902 

32 

29 

.82  112 

.87  457 

.94655 

.05  345 

.12  543 

.17  888 

31 

30 

9.82  126 

9.87  446 

9.94  681 

0.05  319 

0.12  554 

0.17  874 

30 

31 

.82  141 

.87  434 

.94706 

.05  294 

.12  566 

.17  859 

29 

32 

.82  155 

.87  423 

.94  732 

.05  268 

.12  577 

.17845 

28 

33 

.82  169 

.87  412 

.94  757 

.05  243 

.12  588 

.17831 

27 

34 

.82184 

.87  401 

.94  783 

•05  217 

.12  599 

.17  816 

26 

35 

9.82  198 

9.87  390 

9.94  808 

0.05  192 

0.12610 

0.17  802 

25 

36 

.82  212 

.87  378 

.94834 

.05  166 

.12  622 

.17  788 

24 

37 

.82  226 

.87  367 

.94  859 

.05  141 

.12  633 

.17  774 

23 

38 

.82  240 

.87  356 

.94884 

.05  116 

.12644 

.17  760 

22 

39 

.82  255 

.87  345 

.94  910 

.05  090 

.12  655 

.17  745 

21 

40 

9.82  269 

9.87  334 

9.94  935 

0.05  065 

0.12  666 

0.17  731 

20 

41 

.82  283 

.87  322 

.94  961 

.05  039 

.12  678 

.17717 

19 

42 

.82  297 

.87311 

.94  986 

.05  014 

.12  689 

.17  703 

18 

43 

.82311 

.87  300 

.95  012 

.04988 

.12  700 

.17  689 

17 

44 

.82  326 

.87  288 

.95  037 

.04963 

.12712 

.17  674 

16 

45 

9.82  340 

9.87  277 

9.95  062 

0.04  938 

0.12  723 

0.17  660 

15 

46 

.82  354 

.87  266 

.95  088 

.04912 

.12  734 

.17  646 

14 

47 

.82  368 

.87  255 

.95  113 

.04887 

.12  745 

.17  632 

13 

48 

•82  382 

.87  243 

.95  139 

.04861 

.12  757 

.17618 

12 

49 

.82  396 

.87  232 

.95  164 

.04836 

.12  768 

.17  604 

11 

50 

9.82  410 

9.87  221 

9.95  190 

0.04  810 

0.12  779 

0.17  590 

10 

51 

.82  424 

.87  209 

.95  215 

.04785 

.12  791 

.17  576 

9 

52 

.82  439 

.87  198 

.95  240 

.04760 

.12  802 

.17561 

8 

53 

.82  453 

.87  187 

.95  266 

.04734 

.12813 

.17  547 

7 

54 

.82  467 

.87  175 

.95  291 

.04709 

.12  825 

.17  533 

6 

55 

9.82  481 

9.87  164 

9.95  317 

0.04  683 

0.12  836 

0.17519 

5 

56 

.82  495 

.87  153 

.95  342 

.04658 

.12847 

.17  505 

4 

57 

.82  509 

.87  141 

.95  368 

.04632 

.12  859 

.17491 

3 

58 

.82  523 

.87  130 

.95  393 

.04607 

.12  870 

.17  477 

2 

59 

.82  537 

.87  119 

.95418 

.04582 

.12  881 

.17  463 

1 

60 

9.82  551 

9.87  107 

9.95  444 

0.04  556 

0.12  893 

0.17449 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

131°  (311°) 


(228°)  48° 


238 


Table  4.    Trigonometric  Logarithms 


42°  (222°) 


(317°)  137C 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.82  551 

9.87  107 

9.95  444 

0.04  556 

0.12  893 

0.17  449 

60 

1 

.82  565 

.87  096 

.95  469 

.04531 

.12  904 

.17  435 

59 

2 

.82  579 

.87  085 

.95  495 

.04  505 

.12915 

.17421 

58 

3 

.82  593 

.87  073 

.95  520 

.04  480 

.12  927 

.17  407 

57 

4 

.82  607 

.87  062 

.95  545 

.04  455 

.12  938 

.17  393 

56 

5 

9.82  621 

9.87  050 

9.95  571 

0.04  429 

0.12  950 

0.17  379 

55 

6 

.82  635 

.87  039 

.95  596 

.04  404 

.12  961 

.17  365 

54 

7 

.82  649 

.87  028 

.95  622 

.04  378 

.12  972 

.17351 

53 

8 

.82  663 

.87  016 

.95  647 

.04  353 

.12  984 

.17  337 

52 

9 

.82  677 

.87  005 

.95  672 

.04  328 

.12  995 

.17  323 

51 

10 

9.82  691 

9.86  993 

9.95  698 

0.04  302 

0.13  007 

0.17  309 

50 

11 

.82  705 

.86  982 

.95  723 

.04  277 

.13018 

.17  295 

49 

12 

.82  719 

.86  970 

.95  748 

.04  252 

.13  030 

.17  281 

48 

13 

.82  733 

.86  959 

.95  774 

.04  226 

.13  041 

.17  267 

47 

14 

.82  747 

.86  947 

.95  799 

.04  201 

.13  053 

.17  253 

46 

15 

9.82  761 

9.86  936 

9.95  825 

0.04  175 

0.13  064 

0.17  239 

45 

16 

.82  775 

.86  924 

.95  850 

.04  150 

.13  076 

.17  225 

44 

17 

.82  788 

.86  913 

.95  875 

.04125 

.13  087 

.17212 

43 

18 

.82  802 

.86  902 

.95  901 

.04  099 

.13  098 

.17  198 

42 

19 

.82  816 

.86  890 

.95  926 

.04  074 

.13  110 

'  .17184 

41 

20 

9.82  830 

9.86  879 

9.95  952 

0.04  048 

0.13  121 

0.17  170 

40 

21 

.82844 

.86  867 

.95  977 

.04  023 

.13  133 

.17  156 

39 

22 

.82  858 

.86  855 

.96  002 

.03  998 

.13  145 

.17  142 

38 

23 

.82  872 

.86844 

.96  028 

.03  972 

.13  156 

.17  128 

37 

24 

.82  885 

.86  832 

.96  053 

.03  947 

.13  168 

.17  115 

36 

25 

9.82  899 

9.86  821 

9.96  078 

0.03  922 

0.13  179 

0.17  101 

35 

26 

.82  913 

.86  809 

.96  104 

.03  896 

.13  191 

.17  087 

34 

27 

.82  927 

.86  798 

.96  129 

.03  871 

.13  202 

.17  073 

33 

28 

.82  941 

.86  786 

.96  155 

.03  845 

.13214 

.17  059 

32 

29 

.82  955 

.86  775 

.96  180 

.03  820 

.13  225 

.17  045 

31 

30 

9.82  968 

9.86  763 

9.96  205 

0.03  795 

0.13  237 

0.17  032 

30 

31 

.82  982 

.86  752 

.96  231 

.03  769 

.13  248 

.17018 

29 

32 

.82  996 

.86  740 

.96  256 

.03  744 

.13  260 

.17  004 

28 

33 

.83  010 

.86  728 

.96  281 

.03  719 

.13  272 

.16  990 

27 

34 

.83  023 

.86  717 

.96  307 

.03  693 

.13  283 

.16  977 

26 

35 

9.83  037 

9.86  705 

9.96  332 

0.03  668 

0.13  295 

0.16  963 

25 

36 

.83  051 

.86  694 

.96  357 

.03  643 

.13  306 

.16  949 

24 

37 

.83  065 

.86  682 

.96  383 

.03  617 

.13318 

.16  935 

23 

38 

.83  078 

.86  670 

.96  408 

.03  592 

.13  330 

.16  922 

22 

39 

.83  092 

.86  659 

.96  433 

.03  567 

.13  341 

.16  908 

21 

40 

9.83  106 

9.86  647 

9.96  459 

0.03  541 

0.13  353 

0.16  894 

20 

41 

.83  120 

.86  635 

.96  484 

.03  516 

.13  365 

.16  880 

19 

42 

.83  133 

.86  624 

.96  510 

.03  490 

.13  376 

.16  867 

18 

43 

.83  147 

.86  612 

.96  535 

.03  465 

.13  388 

.16  853 

17 

44 

.83  161 

.86  600 

.96  560 

.03  440 

.13  400 

.16  839 

16 

45 

9.83  174 

9.86  589 

9.96  586 

0.03  414 

0.13411 

0.16  826 

15 

46 

.83  188 

.86  577 

.96611 

.03  389 

.13  423 

.16812 

14 

47 

.83  202 

.86  565 

.96  636 

.03  364 

.13435 

.16  798 

13 

48 

.83  215 

.86  554 

.96  662 

.03  338 

.13  446 

.16  785 

12 

49 

.83  229 

.86  542 

.96  687 

.03  313 

.13  458 

.16771 

11 

50 

9.83  242 

9.86  530 

9.96  712 

0.03  288 

0.13  470 

0.16  758 

10 

51 

.83  256 

.86  518 

.96  738 

.03  262 

.13  482 

.16  744 

9 

52 

.83  270 

.86  507 

.96  763 

.03  237 

.13  493 

.16  730 

8 

53 

.83  283 

.86  495 

.96  788 

.03  212 

.13  505 

.16717 

7 

54 

.83  297 

.86  483 

.96  814 

.03  186 

.13517 

.16  703 

6 

55 

9.83  310 

9.86  472 

9.96  839 

0.03  161 

0.13  528 

0.16  690 

5 

56 

.83  324 

.86  460 

.96  864 

.03  136 

.13  540 

.16  676 

4 

57 

.83  338 

.86  448 

.96  890 

.03  110 

.13  552 

.16  662 

3 

58 

.83  351 

.86  436 

.96  915 

.03  085 

.13  564 

.16  649 

2 

59 

.83  365 

.86  425 

.96  940 

.03  060 

.13  575 

.16  635 

1 

60 

9.83  378 

9.86  413 

9.96  966 

0.03  034 

0.13  587 

0.16  622 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

132°  (312°) 


(227°)  47° 


Table  4.    Trigonometric  Logarithms 


239 


43°  (223°) 


(316°)  136° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.83  378 

9.86413 

9.96  966 

0.03  034 

0.13  587 

0.16622 

60 

1 

.83  392 

.86  401 

.96  991 

.03  009 

.13  599 

.16  608 

59 

2 

.83  405 

.86  389 

.97  016 

.02  984 

.13611 

.16  595 

58 

3 

.83419 

.86  377 

.97  042 

.02  958 

.13  623 

.16  581 

57 

4 

.83  432 

.86  366 

.97  067 

.02  933 

.13  634 

.16  568 

56 

5 

9.83  446 

9.86  354 

9.97  092 

0.02  908 

0.13  646 

0.16  554 

55 

6 

.83459 

.86  342 

.97  118 

.02  882 

.13  658 

.16541 

54 

7 

.83  473 

.86  330 

.97  143 

.02  857 

.13  670 

.16  527 

53 

8 

.83  486 

.86  318 

.97  168 

.02  832 

.13  682 

.16514 

52 

9 

.83500 

.86  306 

.97  193 

.02  807 

.13  694 

.16  500 

51 

10 

9.83  513 

9.86  295 

9.97  219 

0.02  781 

0.13  705 

0.16  487 

50 

11 

.83  527 

.86  283 

.97  244 

.02  756 

.13717 

.16  473 

49 

12 

.83  540 

.86  271 

.97  269 

.02  731 

.13  729 

.16  460 

48 

13 

.83  554 

.86  259 

.97  295 

.02  705 

.13  741 

.16  446 

47 

14 

.83567 

.86  247 

.97  320 

.02  680 

.13  753 

.16  433 

46 

15 

9.83  581 

9.86  235 

9.97  345 

0.02  655 

0.13  765 

0.16419 

45 

16 

.83594 

.86  223 

.97  371 

.02  629 

.13  777 

.16  406 

44 

17 

.83  608 

.86211 

.97  396 

.02  604 

.13  789 

.16  392 

43 

18 

.83  621 

.86  200 

.97  421 

.02  579 

.13  800 

.16  379 

42 

19 

.83  634 

.86  188 

.97  447 

.02  553 

.13812 

.16  366 

41 

20 

9.83  648 

9.86  176 

9.97  472 

0.02  528 

0.13  824 

0.16  352 

40 

21 

.83  661 

.86  164 

.97  497 

.02  503 

.13  836 

.16  339 

39 

22 

.83  674 

.86  152 

.97  523 

.02  477 

.13848 

.16326 

38 

23 

.83  688 

.86  140 

.97  548 

.02  452 

.13  860 

.16312 

37 

24 

.83  701 

.86  128 

.97  573 

.02427 

.13  872 

.16  299 

36 

25 

9.83  715 

9.86  116 

9.97  598 

0.02  402 

0.13  884 

0.16  285 

35 

26 

.83  728 

.86  104 

.97  624 

.02  376 

.13  896 

.16  272 

34 

27 

.83741 

.86  092 

.97  649 

.02  351 

.13  908 

.16  259 

33 

28 

.83  755 

.86  080 

.97  674 

.02  326 

.13  920 

.16  245 

32 

29 

.83  768 

.86  068 

.97  700 

.02  300 

.13  932 

.16  232 

31 

30 

9.83  781 

9.86  056 

9.97  725 

0.02  275 

0.13  944 

0.16219 

30 

31 

.83  795 

.86  044 

.97  750 

.02  250 

.13  956 

.16  205 

29 

32 

.83  808 

.86  032 

.97  776 

.02  224 

.13  968 

.16  192 

28 

33 

.83  821 

.86  020 

.97  801 

.02  199 

.13  980 

.16  179 

27 

34 

.83834 

.86  008 

.97  826 

.02  174 

.13  992 

.16  166 

26 

35 

9.83  848 

9.85  996 

9.97  851 

0.02  149 

0.14  004 

0.16  152 

25 

36 

.83  861 

.85  984 

.97  877 

.02  123 

.14  016 

.16  139 

24 

37 

.83  874 

.85  972 

.97  902 

.02  098 

.14  028 

.16  126 

23 

38 

.83  887 

.85  960 

.97  927 

.02  073 

.14  040 

.16  113 

22 

39 

.83  901 

.85948 

.97  953 

.02  047 

.14  052 

.16  099 

21 

40 

9.83  914 

9.85  936 

9.97  978 

0.02  022 

0.14  064 

0.16  086 

20 

41 

.83  927 

.85  924 

.98  003 

.01  997 

.14  076 

.16  073 

19 

42 

.83  940 

.85  912 

.98  029 

.01  971 

.14  088 

.16  060 

18 

43 

.83  954 

.85  900 

.98  054 

.01  946 

.14  100 

.16  046 

17 

44 

.83  967 

.85  888 

.98  079 

.01  921 

.14  112 

.16  033 

16 

45 

9.83  980 

9.85  876 

9.98  104 

0.01  896 

0.14  124 

0.16  020 

15 

46 

.83  993 

.85864 

.98  130 

.01  870 

.14  136 

.16  007 

14 

47 

.84006 

.85851 

.98  155 

.01845 

.14  149 

.15  994 

13 

48 

.84020 

.85  839 

.98  180 

.01  820 

.14  161 

.15  980 

12 

49 

.84033 

.85  827 

.98  206 

.01  794 

.14  173 

.15  967 

11 

50 

9.84046 

9.85  815 

9.98  231 

0.01  769 

0.14  185 

0.15  954 

10 

51 

.84059 

.85  803 

.98  256 

.01  744 

.14  197 

.15  941 

9 

52 

.84072 

.85  791 

.98  281 

.01  719 

.14  209 

.15  928 

8 

53 

.84085 

.85  779 

.98  307 

.01  693 

.14  221 

.15915 

7 

54 

.84098 

.85  766 

.98  332 

.01  668 

.14  234 

.15  902 

6 

55 

9.84112 

9.85  754 

9.98  357 

0.01  643 

0.14  246 

0.15  888 

5 

56 

.84125 

.85742 

.98  383 

.01  617 

.14  258 

.15  875 

4 

57 

.84  138 

.85  730 

.98  408 

.01  592 

.14  270 

.15  862 

3 

58 

.84151 

.85  718 

.98  433 

.01  567 

.14  282 

.15849 

2 

59 

.84  164 

.85  706 

.98  458 

.01  542 

.14  294 

.15  836 

1 

60 

9.84  177 

9.85  693 

9.98  484 

0.01  516 

0.14  307 

0.15  823 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

133°  (313°) 


(226°)  46C 


240 


Table  4.    Trigonometric  Logarithms 


44°  (224°) 


(315°)  135= 


Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.84  177 

9.85  693 

9.98  484 

0.01  516 

0.14307 

0.15  823 

60 

1 

.84  190 

.85  681 

.98  509 

.01  491 

.14319 

.15810 

59 

2 

.84203 

.85  669 

.98  534 

.01  466 

.14331 

.15  797 

58 

3 

.84216 

.85  657 

.98  560 

.01  440 

.14  343 

.15  784 

57 

4 

.84229 

.85  645 

.98  585 

.01  415 

.14  355 

.15  771 

56 

5 

9.84  242 

9.85  632 

9.98610 

0.01  390 

0.14  368 

0.15  758 

55 

6 

.84  255 

.85  620 

.98  635 

.01  365 

.14  380 

.15  745 

54 

7 

.84  269 

.85  608 

.98  661 

.01  339 

.14  392 

.15  731 

53 

8 

.84282 

.85  596 

.98  686 

.01  314 

.14404 

.15718 

52 

9 

.84295 

.85  583 

.98  711 

.01  289 

.14417 

.15  705 

51 

10 

9.84  308 

9.85  571 

9.98  737 

0.01  263 

0.14429 

0.15  692 

50 

11 

.84  321 

.85  559 

.98  762 

.01  238 

.14  441 

.15  679 

49 

12 

.84334 

.85  547 

.98  787 

.01  213 

.14  453 

.15  666 

48 

13 

.84347 

.85  534 

.98  812 

.01  188 

.14  466 

.15  653 

47 

14 

.84  360 

.85  522 

.98  838 

.01  162 

.14  478 

.15  640 

46 

15 

9.84  373 

9.85  510 

9.98  863 

0.01  137 

0.14490 

0.15  627 

45 

16 

.84385 

.85  497 

.98  888 

.01  112 

.14  503 

.15  615 

44 

17 

.84  398 

.85  485 

.98  913 

.01  087 

.14515 

.15  602 

43 

18 

.84411 

.85  473 

.98  939 

.01  061 

.14  527 

.15  589 

42 

19 

.84  424 

.85  460 

.98  964 

.01  036 

.14  540 

.15  576 

41 

20 

9.84  437 

9.85  448 

9.98  989 

0.01  Oil 

0.14  552 

0.15  563 

40 

21 

.84  450 

.85  436 

.99  015 

.00  985 

.14  564 

.15  550 

39 

22 

.84  463 

.85  423 

.99  040 

.00  960 

.14  577 

.15  537 

38 

23 

.84476 

.85411 

.99  065 

.00  935 

.14  589 

.15  524 

37 

24 

.84  489 

.85399 

.99  090 

.00  910 

.14  601 

.15511 

36 

25 

9.84  502 

9.85  386 

9.99  116 

0.00  884 

0.14  614 

0.15  498 

35 

26 

.84  515 

.85  374 

.99  141 

.00  859 

.14  626 

.15485 

34 

27 

.84528 

.85  361 

.99  166 

.00  834 

.14  639 

.15472 

33 

28 

.84  540 

.85  349 

.99  191 

.00  809 

.14  651 

.15  460 

32 

29 

.84553 

.85  337 

.99  217 

.00783 

.14  663 

.15447 

31 

30 

9.84  566 

9.85  324 

9.99  242 

0.00  758 

0.14  676 

0.15  434 

30 

31 

.84579 

.85  312 

.99  267 

.00  733 

.14  688 

.15421 

29 

32 

.84592 

.85  299 

.99  293 

.00  707 

.14  701 

.15  408 

28 

33 

.84  605 

.85  287 

.99  318 

.00682 

.14713 

.15  395 

27 

34 

.84  618 

.85  274 

.99  343 

.00657 

.14  726 

.15  382 

26 

35 

9.84  630 

9.85  262 

9.99  368 

0.00  632 

0.14  738 

0.15  370 

25 

36 

.84643 

.85  250 

.99  394 

.00606 

.14  750 

.15  357 

24 

37 

.84  656 

.85237 

.99  419 

.00581 

.14  763 

.15  344 

23 

38 

.84669 

.85  225 

.99  444 

.00  556 

.14  775 

.15331 

22 

39 

.84682 

.85  212 

.99  469 

.00  531 

.14  788 

.15318 

21 

40 

9.84  694 

9.85  200 

9.99  495 

0.00  505 

0.14  800 

0.15306 

20 

41 

.84  707 

.85  187 

.99  520 

.00  480 

.14813 

.15  293 

19 

42 

.84  720 

.85  175 

.99  545 

.00455 

.14  825 

.15  280 

18 

43 

.84733 

.85  162 

.99  570 

.00430 

.14  838 

.15  267 

17 

44 

.84745 

.85  150 

.99  596 

.00  404 

.14  850 

.15  255 

16 

45 

9.84  758 

9.85  137 

9.99  621 

0.00  379 

0.14  863 

0.15  242 

15 

46 

.84771 

.85  125 

.99  646 

.00354 

.14  875 

.15  229 

14 

47 

.84784 

.85  112 

.99  672 

.00  328 

.14  888 

.15216 

13 

48 

.84  796 

.85100 

.99  697 

.00303 

.14  900 

.15  204 

12 

49 

.84809 

.85087 

.99  722 

.00278 

.14  913 

.15  191 

11 

50 

9.84  822 

9.85  074 

9.99  747 

0.00  253 

0.14  926 

0.15  178 

10 

51 

.84835 

.85  062 

.99  773 

.00  227 

.14  938 

.15  165 

9 

52 

.84  847 

.85049 

.99  798 

.00202 

.14951 

.15  153 

8 

53 

.84860 

.85  037 

.99  823 

.00  177 

.14  963 

.15  140 

7 

54 

.84873 

.85  024 

.99848 

.00  152 

.14  976 

.15  127 

6 

55 

9.84  885 

9.85012 

9.99  874 

0.00  126 

0.14  988 

0.15  115 

5 

56 

.84898 

.84999 

.99  899 

.00  101 

.15  001 

.15  102 

4 

57 

.84911 

.84  986 

.99  924 

.00  076 

.15014 

.15  089 

3 

58 

.84923 

.84  974 

.99  949 

.00  051 

.15  026 

.15  077 

2 

59 

.84  936 

.84  961 

.99  975 

.00  025 

.15  039 

.15  064 

1 

60 

9.84  949 

9.84  949 

0.00  000 

0.00  000 

0.15  051 

0.15  051 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

134°  (314°) 


(225°)  45° 


Table  5.    Meridional  Parts 


241 


0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

9° 

0 

0.0 

59.6 

119.2 

178.9 

238.6 

298.3 

358.2 

418.2 

478.3 

538.6 

0 

1 

1.0 

60.6 

20.2 

79.9 

39.6 

99.3 

59.2 

19.2 

79.3 

39.6 

1 

2 

2.0 

61.6 

21.2 

80.8 

40.6 

300.3 

60.2 

20.2 

80.3 

40.6 

2 

3 

3.0 

62.6 

22.2 

81.8 

41.6 

01.3 

61.2 

21.2 

81.3 

41.6 

3 

4 

4.0 

63.6 

23.2 

82.8 

42.5 

02.3 

62.2 

22.2 

82.3 

42.6 

4 

5 

5.0 

64.6 

124.2 

183.8 

243.5 

303.3 

363.2 

423.2 

483.3 

543.6 

5 

6 

6.0 

65.6 

25.2 

84.8 

44.5 

04.3 

64.2 

24.2 

84.3 

44.6 

6 

7 

7.0 

66.5 

26.2 

85.8 

45.5 

05.3 

65.2 

25.2 

85.3 

45.6 

7 

8 

7.9 

67.5 

27.2 

86.8 

46.5 

06.3 

66.2 

26.2 

86.3 

46.6 

8 

9 

8.9 

68.5 

28.2 

87.8 

47.5 

07.3 

67.2 

27.2 

87.3 

47.6 

9 

10 

9.9 

69.5 

129.1 

188.8 

248.5 

308.3 

368.2 

428.2 

488.3 

548.6 

10 

11 

10.9 

70.5 

30.1 

89.8 

49.5 

09.3 

69.2 

29.2 

89.3 

49.6 

11 

12 

11.9 

71.5 

31.1 

90.8 

50.5 

10.3 

70.2 

30.2 

90.4 

50.6 

12 

13 

12.9 

72.5 

32.1 

91.8 

51.5 

11.3 

71.2 

31.2 

91.4 

51.7 

13 

14 

13.9 

73.5 

33.1 

92.8 

52.5 

12.3 

72.2 

32.2 

92.4 

52.7 

14 

15 

14.9 

74.5 

134.1 

193.8 

253.5 

313.3 

373.2 

433.2 

493.4 

553.7 

15 

16 

15.9 

75.5 

35.1 

94.8 

54.5 

14.3 

74.2 

34.2 

94.4 

54.7 

16 

17 

16.9 

76.5 

36.1 

95.8 

55.5 

15.3 

75.2 

35.2 

95.4 

55.7 

17 

18 

17.9 

77.5 

37.1 

96.8 

56.5 

16.3 

76.2 

36.2 

96.4 

56.7 

18 

19 

18.9 

78.5 

38.1 

97.8 

57.5 

17.3 

77.2 

37.2 

97.4 

57.7 

19 

20 

19.9 

79.5 

139.1 

198.8 

258.5 

318.3 

378.2 

438.2 

498.4 

558.7 

20 

21 

20.9 

80.5 

40.1 

99.7 

59.5 

19.3 

79.2 

39.2 

99.4 

59.7 

21 

22 

21.9 

81.5 

41.1 

200.7 

60.5 

20.3 

80.2 

40.2 

500.4 

60.7 

22 

23 

22.8 

82.4 

42.1 

01.7 

61.5 

21.3 

81.2 

41.2 

01.4 

61.7 

23 

24 

23.8 

83.4 

43.1 

02.7 

62.5 

22.3 

82.2 

42.2 

02.4 

62.7 

24 

25 

24.8 

84.4 

144.1 

203.7 

263.5 

323.3 

383.2 

443.2 

503.4 

563.7 

25 

26 

25.8 

85.4 

45.1 

04.7 

64.5 

24.3 

84.2 

44.2 

04.4 

64.7 

26 

27 

26.8 

86.4 

46.0 

05.7 

65.5 

25.3 

85.2 

45.2 

05.4 

65.7 

27 

28 

27.8 

87.4 

47.0 

06.7 

66.5 

26.3 

86.2 

46.2 

06.4 

66.8 

28 

29 

28.8 

88.4 

48.0 

07.7 

67.4 

27.3 

87.2 

47.2 

07.4 

67.8 

29 

30 

29.8 

89.4 

149.0 

208.7 

268.4 

328.3 

388.2 

448.2 

508.4 

568.8 

30 

31 

30.8 

90.4 

50.0 

09.7 

69.4 

29.3 

89.2 

49.2 

09.4 

69.8 

31 

32 

31.8 

91.4 

51.0 

10.7 

70.4 

30.3 

90.2 

50.2 

10.4 

70.8 

32 

33 

32.8 

92.4 

52.0 

11.7 

71.4 

31.3 

91.2 

51.2 

11.4 

71.8 

33 

34 

33.8 

93.4 

53.0 

12.7 

72.4 

32.3 

92.2 

52.2 

12.4 

72.8 

34 

35 

34.8 

94.4 

154.0 

213.7 

273.4 

333.3 

393.2 

453.2 

513.4 

573.8 

35 

36 

35.8 

95.4 

55.0 

14.7 

74.4 

34.3 

94.2 

54.3 

14.5 

74.8 

36 

37 

36.7 

96.4 

56.0 

15.7 

75.4 

35.3 

95.2 

55.3 

15.5 

75.8 

37 

38 

37.7 

97.3 

57.0 

16.7 

76.4 

36.2 

96.2 

56.3 

16.5 

76.8 

38 

39 

38.7 

98.3 

58.0 

17.7 

77.4 

37.2 

97.2 

57.3 

17.5 

77.8 

39 

40 

39.7 

99.3 

159.0 

218.7 

278.4 

338.2 

398.2 

458.3 

518.5 

578.8 

40 

41 

40.7 

100.3 

60.0 

19.7 

79.4 

39.2 

99.2 

59.3 

19.5 

79.9 

41 

42 

41.7 

01.3 

61.0 

20.6 

80.4 

40.2 

400.2 

60.3 

20.5 

80.9 

42 

43 

42.7 

02.3 

62.0 

21.6 

81.4 

41.2 

01.2 

61.3 

21.5 

81.9 

43 

44 

43.7 

03.3 

63.0 

22.6 

82.4 

42.2 

02.2 

62.3 

22.5 

82.9 

44 

45 

44.7 

104.3 

164.0 

223.6 

283.4 

343.2 

403.2 

463.3 

523.5 

583.9 

45 

46 

45.7 

05.3 

65.0 

24.6 

84.4 

44.2 

04.2 

64.3 

24.5 

84.9 

46 

47 

46.7 

06.3 

66.0 

25.6 

85.4 

45.2 

05.2 

65.3 

25.5 

85.9 

47 

48 

47.7 

07.3 

67.0 

26.6 

86.4 

46.2 

06.2 

66.3 

26.5 

86.9 

48 

49 

48.7 

08.3 

68.0 

27.6 

87.4 

47.2 

07.2 

67.3 

27.5 

87.9 

49 

50 

49.7 

109.3 

168.9 

228.6 

288.4 

348.2 

408.2 

468.3 

528.5 

588.9 

50 

51 

50.7 

10.3 

69.9 

29.6 

89.4 

49.2 

09.2 

69.3 

29.5 

89.9 

51 

52 

51.6 

11.3 

70.9 

30.6 

90.4 

50.2 

10.2 

70.3 

30.5 

90.9 

52 

53 

52.6 

12.3 

71.9 

31.6 

91.4 

51.2 

11.2 

71.3 

31.5 

91.9 

53 

54 

53.6 

13.2 

72.9 

32.6 

92.4 

52.2 

12.2 

72.3 

32.5 

93.0 

54 

55 

54.6 

114.2 

173.9 

233.6 

293.4 

353.2 

413.2 

473.3 

533.5 

594.0 

55 

56 

55.6 

15.2 

74.9 

34.6 

94.4 

54.2 

14.2 

74.3 

34.6 

95.0 

56 

57 

56.6 

16.2 

75.9 

35.6 

95.4 

55.2 

15.2 

75.3 

35.6 

96.0 

57 

58 

57.6 

17.2 

76.9 

36.6 

96.3 

56.2 

16.2 

76.3 

36.6 

97.0 

58 

59 

58.6 

18.2 

77.9 

37.6 

97.3 

57.2 

17.2 

77.3 

37.6 

98.0 

59 

60 

59.6 

119.2 

178.9 

238.6 

298.3 

358.2 

418.2 

478.3 

538.6 

599.0 

60 

' 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

' 

242 


Table  5.    Meridional  Parts 


' 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

' 

0 

599.0 

659.6 

720.5 

781.5 

842.8 

904.4 

966.3 

1028.5 

1091.0 

1153.9 

0 

1 

600.0 

60.6 

21.5 

82.5 

43.9 

05.4 

67.3 

29.5 

92.0 

54.9 

1 

2 

01.0 

61.7 

22.5 

83.6 

44.9 

06.5 

68.3 

30.5 

93.1 

56.0 

2 

3 

02.0 

62.7 

23.5 

84.6 

45.9 

07.5 

69.4 

31.6 

94.1 

57.0 

!  3 

4 

03.0 

63.7 

24.5 

85.6 

46.9 

08.5 

70.4 

32.6 

95.2 

58.1 

4 

5 

604.1 

664.7 

725.5 

786.6 

847.9 

909.6 

971.4 

1033.7 

1096.2 

1159.1 

5 

6 

05.1 

65.7 

26.6 

87.6 

49.0 

10.6 

72.5 

34.7 

97.3 

60.2 

6 

7 

06.1 

66.7 

27.6 

88.7 

50.0 

11.6 

73.5 

35.7 

98.3 

61.2 

7 

8 

07.1 

67.7 

28.6 

89.7 

51.0 

12.6 

74.6 

36.8 

99.4 

62.3 

8 

9 

08.1 

68.7 

29.6 

90.7 

52.0 

13.7 

75.6 

37.8 

1100.4 

63.3 

9 

10 

609.1 

669.8 

730.6 

791.7 

853.1 

914.7 

976.6 

1038.9 

1101.4 

1164.4 

10 

11 

10.1 

70.8 

31.6 

92.7 

54.1 

15.7 

77.7 

39.9 

02.5 

65.4 

11 

12 

11.1 

71.8 

32.7 

93.8 

55.1 

16.8 

78.7 

40.9 

03.5 

66.5 

12 

13 

12.1 

72.8 

33.7 

94.8 

56.1 

17.8 

79.7 

42.0 

04.6 

67.5 

13 

14 

13.1 

73.8 

34.7 

95.8 

57.2 

18.8 

80.8 

43.0 

05.6 

68.6 

14 

15 

614.1 

674.8 

735.7 

796.8 

858.2 

919.8 

981.8 

1044.1 

1106.7 

1169.7 

15 

16 

15.2 

75.8 

36.7 

97.8 

59.2 

20.9 

82.8 

45.1 

07.7 

70.7 

16 

17 

16.2 

76.8 

37.7 

98.9 

60.2 

21.9 

83.9 

46.1 

08.8 

71.8 

17 

18 

17.2 

77.9 

38.8 

99.9 

61.3 

22.9 

84.9 

47.2 

09.8 

72.8 

18 

19 

18.2 

78.9 

39.8 

800.9 

62.3 

24.0 

85.9 

48.2 

10.9 

73.9 

19 

20 

619.2 

679.9 

740.8 

801.9 

863.3 

925.0 

987.0 

1049.3 

1111.9 

1174.9 

20 

21 

20.2 

80.9 

41.8 

02.9 

64.3 

26.0 

88.0 

50.3 

13.0 

76.0 

21 

22 

21.2 

81.9 

42.8 

04.0 

65.4 

27.1 

89.0 

51.3 

14.0 

77.0 

22 

23 

22.2 

82.9 

43.8 

05.0 

66.4 

28.1 

90.1 

52.4 

15.0 

78.1 

23 

24 

23.2 

83.9 

44.9 

06.0 

67.4 

29.1 

91.1 

53.4 

16.1 

79.1 

24 

25 

624.2 

684.9 

745.9 

807.0 

868.5 

930.1 

992.1 

1054.5 

1117.1 

1180.2 

25 

26 

25.3 

86.0 

46.9 

08.1 

69.5 

31.2 

93.2 

55.5 

18.2 

81.2 

26 

27 

26.3 

87.0 

47.9 

09.1 

70.5 

32.2 

94.2 

56.6 

19.2 

82.3 

27 

28 

27.3 

88.0 

48.9 

10.1 

71.5 

33.2 

95.3 

57.6 

20.3 

83.3 

28 

29 

28.3 

89.0 

49.9 

11.1 

72.6 

34.3 

96.3 

58.6 

21.3 

84.4 

29 

30 

629.3 

690.0 

751.0 

812.1 

873.6 

935.3 

997.3 

1059.7 

1122.4 

1185.5 

30 

31 

30.3 

91.0 

52.0 

13.2 

74.6 

36.3 

98.4 

60.7 

23.4 

86.5 

31 

32 

31.3 

92.0 

53.0 

14.2 

75.6 

37.4 

99.4 

61.8 

24.5 

87.6 

32 

33 

32.3 

93. 

54.0 

15.2 

76.7 

38.4 

1000.4 

62.8 

25.5 

88.6 

33 

34 

33.3 

94. 

55.0 

16.2 

77.7 

39.4 

01.5 

63.9 

26.6 

89.7 

34 

35 

634.3 

695. 

756.0 

817.3 

878.7 

940.5 

1002.5 

1064.9 

1127.6 

1190.7 

35 

36 

35.4 

96. 

57.1 

18.3 

79.7 

41.5 

03.6 

65.9 

28.7 

91.8 

36 

37 

36.4 

97. 

58.1 

19.3 

80.8 

42.5 

04.6 

67.0 

29.7 

92.8 

37 

38 

37.4 

98.1 

59.1 

20.3 

81.8 

43.6 

05.6 

68.0 

30.8 

93.9 

38 

39 

38.4 

99.1 

60.1 

21.3 

82.8 

44.6 

06.7 

69.1 

31.8 

95.0 

39 

40 

639.4 

700.2 

761.1 

822.4 

883.8 

945.6 

1007.7 

1070.1 

1132.9 

1196.0 

40 

41 

40.4 

01.2 

62.2 

23.4 

84.9 

46.7 

08.7 

71.2 

33.9 

97.1 

41 

42 

41.4 

02.2 

63.2 

24.4 

85.9 

47.7 

09.8 

72.2 

35.0 

98.1 

42 

43 

42.4 

03.2 

64.2 

25.4 

86.9 

48.7 

10.8 

73.2 

36.0 

99.2 

43 

44 

43.4 

04.2 

65.2 

26.5 

88.0 

49.7 

11.8 

74.3 

37.1 

1200.2 

44 

45 

644.5 

705.2 

766.2 

827.5 

889.0 

950.8 

1012.9 

1075.3 

1138.1 

1201.3 

45 

46 

45.5 

06.2 

67.3 

28.5 

90.0 

51.8 

13.9 

76.4 

39.2 

02.3 

46 

47 

46.5 

07.3 

68.3 

29.5 

91.0 

52.8 

15.0 

77.4 

40.2 

03.4 

47 

48 

47.5 

08.3 

69.3 

30.5 

92.1 

53.9 

16.0 

78.5 

41.3 

04.5 

48 

49 

48.5 

09.3 

70.3 

31.6 

93.1 

54.9 

17.0 

79.5 

42.3 

05.5 

49 

50 

649.5 

710.3 

771.3 

832.6 

894.1 

955.9 

1018.1 

1080.5 

1143.4 

1206.6 

50 

51 

50.5 

11.3 

72.3 

33.6 

95.2 

57.0 

19.1 

81.6 

44.4 

07.6 

51 

52 

51.5 

12.3 

73.4 

34.6 

96.2 

58.0 

20.2 

82.6 

45.5 

08.7 

52 

53 

52.5 

13.4 

74.4 

35.7 

97.2 

59.0 

21.2 

83.7 

46.5 

09.7 

53 

54 

53.6 

14.4 

75.4 

36.7 

98.2 

60.1 

22.2 

84.7 

47.6 

10.8 

54 

55 

654.6 

715.4 

776.4 

837.7 

899.3 

961.1 

1023.3 

1085.8 

1148.6 

1211.8 

55 

56 

55.6 

16.4 

77.4 

38.7 

900.3 

62.1 

24.3 

86.8 

49.7 

12.9 

56 

57 

56.6 

17.4 

78.5 

39.8 

01.3 

63.2 

25.3 

87.9 

50.7 

14.0 

57 

58 

57.6 

18.4 

79.5 

40.8 

02.3 

64.2 

26.4 

88.9 

51.8 

15.0 

58 

59 

58.6 

19.4 

80.5 

41.8 

03.4 

65.2 

27.4 

89.9 

52.8 

16.1 

59 

60 

659.6 

720.5 

781.5 

842.8 

904.4 

966.3 

1028.5 

1091.0 

1153.9 

1217.1 

60 

' 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

' 

Table  5.    Meridional  Parts 


243 


1 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

' 

0 

1217.1 

1280.8 

1344.9 

1409.5 

1474.5 

1540.1 

1606.2 

1672.9 

1740.2 

1808.1 

0 

1 

18.2 

81.9 

46.0 

10.6 

75.6 

41.2 

07.3 

74.0 

41.3 

09.2 

1 

2 

19.3 

82.9 

47.1 

11.6 

76.7 

42.3 

08.4 

75.1 

42.4 

10.4 

2 

3 

20.3 

84.0 

48.1 

12.7 

77.8 

43.4 

09.5 

76.2 

43.6 

11.5 

3 

4 

21.4 

85.1 

49.2 

13.8 

78.9 

44.5 

10.6 

77.4 

44.7 

12.6 

4 

5 

1222.4 

1286.1 

1350.3 

1414.9 

1480.0 

1545.6 

1611.7 

1678.5 

1745.8 

1813.8 

5 

6 

23.5 

87.2 

51.4 

16.0 

81.1 

46.7 

12.9 

79.6 

46.9 

14.9 

6 

7 

24.5 

88.3 

52.4 

17.1 

82.2 

47.8 

14.0 

80.7 

48.1 

16.1 

7 

8 

25.6 

89.3 

53.5 

18.1 

83.3 

48.9 

15.1 

81.8 

49.2 

17.2 

8 

9 

26.7 

90.4 

54.6 

19.2 

84.3 

50.0 

16.2 

82.9 

50.3 

18.3 

9 

10 

1227.7 

1291.5 

1355.7 

1420.3 

1485.4 

1551.1 

1617.3 

1684.1 

1751.5 

1819.5 

10 

11 

28.8 

92.5 

56.7 

21.4 

86.5 

52.2 

18.4 

85.2 

52.6 

20.6 

11 

12 

29.8 

93.6 

57.8 

22.5 

87.6 

53.3 

19.5 

86.3 

53.7 

21.8 

12 

13 

30.9 

94.7 

58.9 

23.5 

88.7 

54.4 

20.6 

87.4 

54.8 

22.9 

13 

14 

32.0 

95.7 

59.9 

24.6 

89.8 

55.5 

21.7 

88.5 

56.0 

24.0 

14 

15 

1233.0 

1296.8 

1361.0 

1425.7 

1490.9 

1556.6 

1622.8 

1689.7 

1757.1 

1825.2 

15 

16 

34.1 

97.9 

62.1 

26.8 

92.0 

57.7 

23.9 

90.8 

58.2 

26.3 

16 

17 

35.1 

98.9 

63.2 

27.9 

93.1 

58.8 

25.0 

91.9 

59.4 

27.5 

17 

18 

36.2 

1300.0 

64.2 

29.0 

94.2 

59.9 

26.2 

93.0 

60.5 

28.6 

18 

19 

37.3 

01.1 

65.3 

30.0 

95.2 

61.0 

27.3 

94.1 

61.6 

29.7 

19 

20 

1238.3 

1302.1 

1366.4 

1431.1 

1496.3 

1562.1 

1628.4 

1695.3 

1762.7 

1830.9 

20 

21 

39.4 

03.2 

67.5 

32.2 

97.4 

63.2 

29.5 

96.4 

63.9 

32.0 

21 

22 

40.4 

04.3 

68.5 

33.3 

98.5 

64.3 

30.6 

97.5 

65.0 

33.2 

22 

23 

41.5 

05.3 

69.6 

34.4 

99.6 

65.4 

31.7 

98.6 

66.1 

34.3 

23 

24 

42.6 

06.4 

70.7 

35.4 

1500.7 

66.5 

32.8 

99.7 

67.3 

35.4 

24 

25 

1243.6 

1307.5 

1371.8 

1436.5 

1501.8 

1567.6 

1633.9 

1700.9 

1768.4 

1836.6 

25 

26 

44.7 

08.5 

72.8 

37.6 

02.9 

68.7 

35.0 

02.0 

69.5 

37.7 

26 

27 

45.7 

09.6 

73.9 

38.7 

04.0 

69.8 

36.1 

03.1 

70.7 

38.9 

27 

28 

46.8 

10.7 

75.0 

39.8 

05.1 

70.9 

37.3 

04.2 

71.8 

40.0 

28 

29 

47.9 

11.7 

76.1 

40.9 

06.2 

72.0 

38.4 

05.3 

72.9 

41.2 

29 

30 

1248.9 

1312.8 

1377.1 

1442.0 

1507.3 

1573.1 

1639.5 

1706.5 

1774.1 

1842.3 

30 

31 

50.0 

13.9 

78.2 

43.0 

08.4 

74.2 

40.6 

07.6 

75.2 

43.4 

31 

32 

51.0 

14.9 

79.3 

44.1 

09.4 

75.3 

41.7 

08.7 

76.3 

44.6 

32 

33 

52.1 

16.0 

80.4 

45.2 

10.5 

76.4 

42.8 

09.8 

77.4 

45.7 

33 

34 

53.2 

17.1 

81.5 

46.3 

11.6 

77.5 

43.9 

10.9 

78.6 

46.9 

34 

35 

1254.2 

1318.2 

1382.5 

1447.4 

1512.7 

1578.6 

1645.0 

1712.1 

1779.7 

1848.0 

35 

36 

55.3 

19.2 

83.6 

48.5 

13.8 

79.7 

46.2 

13.2 

80.8 

49.2 

36 

37 

56.4 

20.3 

84.7 

49.5 

14.9 

80.8 

47.3 

14.3 

82.0 

50.3 

37 

38 

57.4 

21.4 

85.8 

50.6 

16.0 

81.9 

48.4 

15.4 

83.1 

51.4 

38 

39 

58.5 

22.4 

86.8 

51.7 

17.1 

83.0 

49.5 

16.6 

84.2 

52.6 

39 

40 

1259.5 

1323.5 

1387.9 

1452.8 

1518.2 

1584.1 

1650.6 

1717.7 

1785.4 

1853.7 

40 

41 

60.6 

24.6 

89.0 

53.9 

19.3 

85.2 

51.7 

18.8 

86.5 

54.9 

41 

42 

61.7 

25.6 

90.1 

55.0 

20.4 

86.3 

52.8 

19.9 

87.6 

56.0 

42 

43 

62.7 

26.7 

91.1 

56.1 

21.5 

87.4 

53.9 

21.1 

88.8 

57.2 

43 

44 

63.8 

27.8 

92.2 

57.1 

22.6 

88.5 

55.1 

22.2 

89.9 

58.3 

44 

45 

1264.9 

1328.9 

1393.3 

1458.2 

1523.7 

1589.6 

1656.2 

1723.3 

1791.1 

1859.5 

45 

46 

65.9 

29.9 

94.4 

59.3 

24.8 

90.7 

57.3 

24.4 

92.2 

60.6 

46 

47 

67.0 

31.0 

95.5 

60.4 

25.9 

91.8 

58.4 

25.5 

93.3 

61.8 

47 

48 

68.0 

32.1 

96.5 

61.5 

27.0 

92.9 

59.5 

26.7 

94.5 

62.9 

48 

49 

69.1 

33.1 

97.6 

62.6 

28.0 

94.1 

60.6 

27.8 

95.6 

64.0 

49 

50 

1270.2 

1334.2 

1398.7 

1463.7 

1529.1 

1595.2 

1661.7 

1728.9 

1796.7 

1865.2 

50 

51 

71.2 

35.3 

99.8 

64.8 

30.2 

96.3 

62.9 

30.0 

97.9 

66.3 

51 

52 

72.3 

36.3 

1400.9 

65.8 

31.3 

97.4 

64.0 

31.2 

99.0 

67.5 

52 

53 

73.4 

37.4 

01.9 

66.9 

32.4 

98.5 

65.1 

32.3 

1800.1 

68.6 

53 

54 

74.4 

38.5 

03.0 

68.0 

33.5 

99.6 

66.2 

33.4 

01.3 

69.8 

54 

55 

1275.5 

1339.6 

1404.1 

1469.1 

1534.6 

1600.7 

1667.3 

1734.5 

1802.4 

1870.9 

55 

56 

76.6 

40.6 

05.2 

70.2 

35.7 

01.8 

68.4 

35.7 

03.5 

72.1 

56 

57 

77.6 

41.7 

06.2 

71.3 

36.8 

02.9 

69.5 

36.8 

04.7 

73.2 

57 

58 

78.7 

42.8 

07.3 

72.4 

37.9 

04.0 

70.7 

37.9 

05.8 

74.4 

58 

59 

79.7 

43.8 

08.4 

73.5 

39.0 

05.1 

71.8 

39.1 

07.0 

75.5 

59 

60 

1280.8 

1344.9 

1409.5 

1474.5 

1540.1 

1606.2 

1672.9 

1740.2 

1808.1 

1876.7 

60 

' 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

' 

244 


Table  5.    Meridional  Parts 


' 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

; 

0 

1876.7 

1946.0 

2016.0 

2086.8 

2158.4 

2230.9 

2304.2 

2378.5 

2453.8 

2530.2 

o 

1 

77.8 

47.1 

17.2 

88.0 

59.6 

32.1 

05.5 

79.8 

55.1 

31.5 

1 

2 

79.0 

48.3 

18.3 

89.2 

60.8 

33.3 

06.7 

81.0 

56.4 

32.8 

2 

3 

80.1 

49.4 

19.5 

90.3 

62.0 

34.5 

07.9 

82.3 

57.6 

34.0 

3 

4 

81.3 

50.6 

20.7 

91.5 

63.2 

35.7 

09.2 

83.5 

58.9 

35.3 

4 

5 

1882.4 

1951.8 

2021.9 

2092.7 

2164.4 

2236.9 

2310.4 

2384.8 

2460.2 

2536.6 

5 

6 

83.6 

52.9 

23.0 

93.9 

65.6 

38.2 

11.6 

86.0 

61.4 

37.9 

6 

7 

84.7 

54.1 

24.2 

95.1 

66.8 

39.4 

12.9 

87.3 

62.7 

39.2 

7 

8 

85.9 

55.3 

25.4 

96.3 

68.0 

40.6 

14.1 

88.5 

64.0 

40.5 

8 

9 

87.0 

56.4 

26.6 

97.5 

69.2 

41.8 

15.3 

89.8 

65.2 

41.7 

9 

10 

1888.2 

1957.6 

2027.7 

2098.7 

2170.4 

2243.0 

2316.5 

2391.0 

2466.5 

2543.0 

10 

11 

89.3 

58.7 

28.9 

99.8 

71.6 

44.2 

17.8 

92.3 

67.8 

44.3 

11 

12 

90.5 

59.9 

30.1 

2101.0 

72.8 

45.5 

19.0 

93.5 

69.0 

45.6 

12 

13 

91.6 

61.1 

31.3 

02.2 

74.0 

46.7 

20.3 

94.8 

70.3 

46.9 

13 

14 

92.8 

62.2 

32.4 

03.4 

75.2 

47.9 

21.5 

96.0 

71.6 

48.2 

14 

15 

1893.9 

1963.4 

2033.6 

2104.6 

2176.4 

2249.1 

2322.7 

2397.3 

2472.8 

2549.5 

15 

16 

95.1 

64.6 

34.8 

05.8 

77.6 

50.3 

24.0 

98.5 

74.1 

50.7 

16 

17 

96.2 

65.7 

36.0 

07.0 

78.8 

51.6 

25.2 

99.8 

75.4 

52.0 

17 

18 

97.4 

66.9 

37.1 

08.2 

80.0 

52.8 

26.4 

2401.0 

76.6 

53.3 

18 

19 

98.5 

68.1 

38.3 

09.4 

81.2 

54.0 

27.7 

02.3 

77.9 

54.6 

19 

20 

1899.7 

1969.2 

2039.5 

2110.6 

2182.5 

2255.2 

2328.9 

2403.5 

2479.2 

2555.9 

20 

21 

1900.8 

70.4 

40.7 

11.8 

83.7 

56.4 

30.1 

04.8 

80.4 

57.2 

21 

22 

02.0 

71.5 

41.8 

12.9 

84.9 

57.7 

31.4 

06.0 

81.7 

58.5 

22 

23 

03.1 

72.7 

43.0 

14.1 

86.1 

58.9 

32.6 

07.3 

83.0 

59.8 

23 

24 

04.3 

73.9 

44.2 

15.3 

87.3 

60.1 

33.8 

08.5 

84.3 

61.0 

24 

25 

1905.5 

1975.0 

2045.4 

2116.5 

2188.5 

2261  3 

2335.1 

2409.8 

2485.5 

2562.3 

25 

26 

06.6 

76.2 

46.6 

17.7 

89.7 

62.5 

36.3 

11.1 

86.8 

63.6 

26 

27 

07.8 

77.4 

47.7 

18.9 

90.9 

63.8 

37.6 

12.3 

88.1 

64.9 

27 

28 

08.9 

78.5 

48.9 

20.1 

92.1 

65.0 

38.8 

13.6 

89.3 

66.2 

28 

29 

10.1 

79.7 

50.1 

21.3 

93.3 

66.2 

40.0 

14.8 

90.6 

67.5 

29 

30 

1911.2 

1980.9 

2051.3 

2122.5 

2194.5 

2267.4 

2341.3 

2416.1 

2491.9 

2568.8 

30 

31 

12.4 

82.0 

52.5 

23.7 

95.7 

68.7 

42.5 

17.3 

93.2 

70.1 

31 

32 

13.5 

83.2 

53.6 

24.9 

96.9 

69.9 

43.7 

18.6 

94.4 

71.4 

32 

33 

14.7 

84.4 

54.8 

26.1 

98.1 

71.1 

45.0 

19.8 

95.7 

72.7 

33 

34 

15.8 

85.5 

56.0 

27.3 

99.4 

72.3 

46.2 

21.1 

97.0 

73.9 

34 

35 

1917.0 

1986.7 

2057.2 

2128.5 

2200.6 

2273.5 

2347.5 

2422.3 

2498.3 

2575.2 

35 

36 

18.2 

87.9 

58.4 

29.6 

01.8 

74.8 

48.7 

23.6 

99.5 

76.5 

36 

37 

19.3 

89.1 

59.5 

30.8 

03.0 

76.0 

49.9 

24.9 

2500.8 

77.8 

37 

38 

20.5 

90.2 

60.7 

32.0 

04.2 

77.2 

51.2 

26.1 

02.1 

79.1 

38 

39 

21.6 

91.4 

61.9 

33.2 

05.4 

78.4 

52.4 

27.4 

03.4 

80.4 

39 

40 

1922.8 

1992.6 

2063.1 

2134.4 

2206.6 

2279.7 

2353.7 

2428.6 

2504.6 

2581.7 

40 

41 

23.9 

93.7 

64.3 

35.6 

07.8 

80.9 

54.9 

29.9 

05.9 

83.0 

41 

42 

25.1 

94.9 

65.5 

36.8 

09.0 

82.1 

56.1 

31.2 

07.2 

84.3 

42 

43 

26.3 

96.1 

66.6 

38.0 

10.2 

83.3 

57.4 

32.4 

08.5 

85.6 

43 

44 

27.4 

97.2 

67.8 

39.2 

11.5 

84.6 

58.6 

33.7 

09.7 

86.9 

44 

45 

1928.6 

1998.4 

2069.0 

2140.4 

2212.7 

2285.8 

2359.9 

2434.9 

2511.0 

2588.2 

45 

46 

29.7 

99.6 

70.2 

41.6 

13.9 

87.0 

61.1 

36.2 

12.3 

89.5 

46 

47 

30.9 

2000.7 

71.4 

42.8 

15.1 

88.3 

62.4 

37.4 

13.6 

90.8 

47 

48 

32.0 

01.9 

72.6 

44.0 

16.3 

89.5 

63.6 

38.7 

14.8 

92.1 

48 

49 

33.2 

03.1 

73.7 

45.2 

17.5 

90.7 

64.8 

40.0 

16.1 

93.4 

49 

50 

1934.4 

2004.3 

2074.9 

2146.4 

2218.7 

2291.9 

2366.1 

2441.2 

2517.4 

2594.7 

50 

51 

35.5 

05.4 

76.1 

47.6 

19.9 

93.2 

67.3 

42.5 

18.7 

96.0 

51 

52 

36.7 

06.6 

77.3 

48.8 

21.1 

94.4 

68.6 

43.7 

20.0 

97.3 

52 

53 

37.8 

07.8 

78.5 

50.0 

22.4 

95.6 

69.8 

45.0 

21.2 

98.5 

53 

54 

39.0 

08.9 

79.7 

51.2 

23.6 

96.9 

71.1 

46.3 

22.5 

99.8 

54 

55 

1940.2 

2010.1 

2080.8 

2152.4 

2224.8 

2298.1 

2372.3 

2447.5 

2523.8 

2601.1 

55 

56 

41.3 

11.3 

82.0 

53.6 

26.0 

99.3 

73.6 

48.8 

25.1 

02.4 

56 

57 

42.5 

12.5 

83.2 

54.8 

27.2 

2300.5 

74.8 

50.1 

26.4 

03.7 

57 

58 

43.6 

13.6 

84.4 

56.0 

28.4 

01.8 

76.1 

51.3 

27.6 

05.0 

58 

59 

44.8 

14.8 

85.6 

57.2 

29.6 

03.0 

77.3 

52.6 

28.9 

06.3 

59 

60 

1946.0 

2016.0 

2086.8 

2158.4 

2230.9 

2304.2 

2378.5 

2453.8 

2530.2 

2607.6 

60 

' 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

' 

Table  5.    Meridional  Parts 


245 


' 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

' 

0 

2607.6 

2686.2 

2766.0 

2847.1 

2929.5 

3013.4 

3098.7 

3185.6 

3274.1 

3364.4 

0 

1 

08.9 

87.6 

67.4 

48.5 

30.9 

14.8 

3100.1 

87.1 

75.6 

65.9 

1 

2 

10.2 

88.9 

68.7 

49.9 

32.3 

16.2 

01.6 

88.5 

77.1 

67.4 

2 

3 

11.5 

90.2 

70.1 

51.2 

33.7 

17.6 

03.0 

90.0 

78.6 

69.0 

3 

4 

12.8 

91.5 

71.4 

52.6 

35.1 

19.0 

04.4 

91.4 

80.1 

70.5 

4 

5 

2614.1 

2692.8 

2772.8 

2853.9 

2936.5 

3020.4 

3105.9 

3192.9 

3281.6 

3372.0 

5 

6 

15.4 

94.2 

74.1 

55.3 

37.9 

21.8 

07.3 

94.4 

83.1 

73.5 

6 

7 

16.8 

95.5 

75.4 

56.7 

39.3 

23.3 

08.8 

95.8 

84.6 

75.1 

7 

8 

18.1 

96.8 

76.8 

58.0 

40.6 

24.7 

10.2 

97.3 

86.1 

76.6 

8 

9 

19.4 

98.1 

78.1 

59.4 

42.0 

26.1 

11.6 

98.8 

87.6 

78.1 

9 

10 

2620.7 

2699.5 

2779.5 

2860.8 

2943.4 

3027.5 

3113.1 

3200.2 

3289.0 

3379.6 

10 

11 

22.0 

2700.8 

80.8 

62.1 

44.8 

28.9 

14.5 

01.7 

90.5 

81.2 

11 

12 

23.3 

02.1 

82.2 

63.5 

46.2 

30.3 

16.0 

03.2 

92.0 

82.7 

12 

13 

24.6 

03.4 

83.5 

64.9 

47.6 

31.7 

17.4 

04.6 

93.5 

84.2 

13 

14 

25.9 

04.8 

84.8 

66.2 

49.0 

33.2 

18.8 

06.1 

95.0 

85.7 

14 

15 

2627.2 

2706.1 

2786.2 

2867.6 

2950.4 

3034,6 

3120.3 

3207.6 

3296.5 

3387.3 

15 

16 

28.5 

07.4 

87.5 

69.0 

51.8 

36.0 

21.7 

09.0 

98.0 

88.8 

16 

17 

29.8 

08.7 

88.9 

70.3 

53.2 

37.4 

23.2 

10.5 

99.5 

90.3 

17 

18 

31.1 

10.1 

90.2 

71.7 

54.5 

38.8 

24.6 

12.0 

3301.0 

91.8 

18 

19 

32.4 

11.4 

91.6 

73.1 

55.9 

40.2 

26.0 

13.4 

02.5 

93.4 

19 

20 

2633.7 

2712.7 

2792.9 

2874.4 

2957.3 

3041.7 

3127.5 

3214.9 

3304.0 

3394.9 

20 

21 

35.0 

14.0 

94.3 

75.8 

58.7 

43.1 

28.9 

16.4 

05.5 

96.4 

21 

22 

36.3 

15.4 

95.6 

77.2 

60.1 

44.5 

30.4 

17.9 

07.0 

98.0 

22 

23 

37.6 

16.7 

97.0 

78.6 

61.5 

45.9 

31.8 

19.3 

08.5 

99.5 

23 

24 

38.9 

18.0 

98.3 

79.9 

62.9 

47.3 

33.3 

20.8 

10.0 

3401.0 

24 

25 

2640.2 

2719.3 

2799.7 

2881.3 

2964.3 

3048.7 

3134.7 

3222.3 

3311.5 

3402.6 

25 

26 

41.6 

20.7 

2801.0 

82.7 

65.7 

50.2 

36.2 

23.7 

13.0 

04.1 

26 

27 

42.9 

22.0 

02.4 

84.0 

67.1 

51.6 

37.6 

25.2 

14.5 

05.6 

27 

28 

44.2 

23.3 

03.7 

85.4 

68.5 

53.0 

39.0 

26.7 

16.0 

07.2 

28 

29 

45.5 

24.7 

05.1 

86.8 

69.9 

54.4 

40.5 

28.2 

17.5 

08.7 

29 

30 

2646.8 

2726.0 

2806.4 

2888.2 

2971.3 

3055.9 

3141.9 

3229.6 

3319.0 

3410.2 

30 

31 

48.1 

27.3 

07.8 

89.5 

72.7 

57.3 

43.4 

31.1 

20.5 

11.8 

31 

32 

49.4 

28.6 

09.1 

90.9 

74.1 

58.7 

44.8 

32.6 

22.1 

13.3 

32 

33 

50.7 

30.0 

10.5 

92.3 

75.5 

60.1 

46.3 

34.1 

23.6 

14.8 

33 

34 

52.0 

31.3 

11.8 

93.7 

76.9 

61.5 

47.7 

35.6 

25.1 

16.4 

34 

35 

2653.3 

2732.6 

2813.2 

2895.0 

2978.3 

3063.0 

3149.2 

3237.0 

3326.6 

3417.9 

35 

36 

54.7 

34.0 

14.5 

96.4 

79.7 

64.4 

50.6 

38.5 

28.1 

19.5 

36 

37 

56.0 

35.3 

15.9 

97.8 

81.1 

65.8 

52.1 

40.0 

29.6 

21.0 

37 

38 

57.3 

36.6 

17.2 

99.2 

82.5 

67.2 

53.5 

41.5 

31.1 

22.5 

38 

39 

58.6 

38.0 

18.6 

2900.5 

83.9 

68.7 

55.0 

42.9 

32.6 

24.1 

39 

40 

2659.9 

2739.3 

2820.0 

2901.9 

2985.3 

3070.1 

3156.4 

3244.4 

3334.1 

3425.6 

40 

41 

61.2 

40.6 

21.3 

03.3 

86.7 

71.5 

57.9 

45.9 

35.6 

27.2 

41 

42 

62.5 

42.0 

22.7 

04.7 

88.1 

72.9 

59.4 

47.4 

37.1 

28.7 

42 

43 

63.9 

43.3 

24.0 

06.1 

89.5 

74.4 

60.8 

48.9 

38.6 

30.2 

43 

44 

65.2 

44.6 

25.4 

07.4 

90.9 

75.8 

62.3 

50.3 

40.2 

31.8 

44 

45 

2666.5 

2746.0 

2826.7 

2908.8 

2992.3 

3077.2 

3163.7 

3251.8 

3341.7 

3433.3 

45 

46 

67.8 

47.3 

28.1 

10.2 

93.7 

78.7 

65.2 

53.3 

43.2 

34.9 

46 

47 

69.1 

48.6 

29.4 

11.6 

95.1 

80.1 

66.6 

54.8 

44.7 

36.4 

47 

48 

70.4 

50.0 

30.8 

13.0 

96.5 

81.5 

68.1 

56.3 

46.2 

38.0 

48 

49 

71.7 

51.3 

32.2 

14.3 

97.9 

82.9 

69.5 

57.8 

47.7 

39.5 

49 

50 

2673.1 

2752.7 

2833.5 

2915.7 

2999.3 

3084.4 

3171.0 

3259.3 

3349.2 

3441.0 

50 

51 

74.4 

54.0 

34.9 

17.1 

3000.7 

85.8 

72.5 

60.7 

50.8 

42.6 

51 

52 

75.7 

55.3 

36.2 

18.5 

02.1 

87.2 

73.9 

62.2 

52.3 

44.1 

52 

53 

77.0 

56.7 

37.6 

19.9 

03.5 

88.7 

75.4 

63.7 

53.8 

45.7 

53 

54 

78.3 

58.0 

39.0 

21.2 

04.9 

90.1 

76.8 

65.2 

55.3 

47.2 

54 

55 

2679.6 

2759.3 

2840.3 

2922.6 

3006.3 

3091.5 

3178.3 

3266.7 

3356.8 

3448.8 

55 

56 

81.0 

60.7 

41.7 

24.0 

07.7 

93.0 

79.7 

68.2 

58.3 

50.3 

56 

57 

82.3 

62.0 

43.0 

25.4 

09.2 

94.4 

81.2 

69.7 

59.9 

51.9 

57 

58 

83.6 

63.4 

44.4 

26.8 

10.6 

95.8 

82.7 

71.1 

61.4 

53.4 

58 

59 

84.9 

64.7 

45.8 

28.2 

12.0 

97.3 

84.1 

72.6 

62.9 

55.0 

59 

60 

2686.2 

2766.0 

2847.1 

2929.5 

3013.4 

3098.7 

3185.6 

3274.1 

3364.4 

3456.5 

60 

' 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

/ 

246 


Table  5.    Meridional  Parts 


/ 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

/ 

0 

3456.5 

3550.6 

3646.7 

3745.1 

3845.7 

3948.8 

4054.5 

4163.0 

4274.4 

4389.1 

0 

1 

58.1 

52.2 

48.4 

46.7 

47.4 

50.5 

56.3 

64.8 

76.3 

91.0 

1 

2 

59.6 

53.8 

50.0 

48.4 

49.1 

52.3 

58.1 

66.6 

78.2 

92.9 

2 

3 

61.2 

55.4 

51.6 

50.0 

50.8 

54.0 

59.8 

68.5 

80.1 

94.9 

3 

4 

62.7 

56.9 

53.2 

51.7 

52.5 

55.7 

61.6 

70.3 

82.0 

96.8 

4 

5 

464.3 

558.5 

3654.8 

3753.4 

3854.2 

3957.5 

4063.4 

4172.1 

4283.9 

4398.8 

5 

6 

65.9 

60.1 

56.5 

55.0 

55.9 

59.2 

65.2 

74.0 

85.7 

4400.7 

6 

'  7 

67.4 

61.7 

58.1 

56.7 

57.6 

61.0 

67.0 

75.8 

87.6 

02.6 

7 

8 

69.0 

63.3 

59.7 

58.3 

59.3 

62.7 

68.8 

77.7 

89.5 

04.6 

8 

9 

70.5 

64.9 

61.3 

60.0 

61.0 

64.5 

70.6 

79.5 

91.4 

06.5 

9 

10 

3472.1 

566.5 

3663.0 

3761.7 

3862.7 

3966.2 

4072.4 

4181.3 

4293.3 

4408.5 

10 

11 

73.6 

68.1 

64.6 

63.3 

64.4 

68.0 

74.2 

83.2 

95.2 

10.4 

11 

12 

75.2 

69.7 

66.2 

65.0 

66.1 

69.7 

76.0 

85.0 

97.1 

12.4 

12 

13 

76.7 

71.3 

67.9 

66.7 

67.8 

71.5 

77.7 

86.9 

99.0 

14.3 

13 

14 

78.3 

72.8 

69.5 

68.3 

69.5 

73.2 

79.5 

88.7 

4300.9 

16.3 

14 

15 

3479.9 

3574.4 

3671.1 

3770.0 

3871.2 

3975.0 

4081.3 

4190.6 

4302.8 

4418.2 

15 

16 

81.4 

76.0 

72.7 

71.7 

72.9 

76.7 

83.1 

92.4 

04.7 

20.2 

16 

17 

83.0 

77.6 

74.4 

73.3 

74.6 

78.5 

84.9 

94.2 

06.6 

22.1 

17 

18 

84.5 

79.2 

76.0 

75.0 

76.3 

80.2 

86.7 

96.1 

08.5 

24.1 

18 

19 

86.1 

80.8 

77.6 

76.7 

78.1 

82.0 

88.5 

97.9 

10.4 

26.1 

19 

20 

3487.7 

3582.4 

3679.3 

3778.3 

3879.8 

3983.7 

4090.3 

4199.8 

4312.3 

4428.0 

20 

21 

89.2 

84.0 

80.9 

80.0 

81.5 

85.5 

92.1 

4201.6 

14.2 

30.0 

21 

22 

90.8 

85.6 

82.5 

81.7 

83.2 

87.2 

93.9 

03.5 

16.1 

31.9 

22 

23 

92.4 

87.2 

84.2 

83.3 

84.9 

89.0 

95.7 

05.3 

18.0 

33.9 

23 

24 

93.9 

88.8 

85.8 

85.0 

86.6 

90.7 

97.5 

07.2 

19.9 

35.8 

24 

25 

3495.5 

3590.4 

3687.4 

3786.7 

3888.3 

3992.5 

4099.3 

4209.0 

4321.8 

4437.8 

25 

26 

97.1 

92.0 

89.1 

88.4 

90.0 

94.3 

4101.1 

10.9 

23.7 

39.8 

26 

27 

98.6 

93.6 

90.7 

90.0 

91.8 

96.0 

02.9 

12.8 

25.6 

41.7 

27 

28 

3500.2 

95.2 

92.3 

91.7 

93.5 

97.8 

04.8 

14.6 

27.5 

43.7 

28 

29 

01.8 

96.8 

94.0 

93.4 

95.2 

99.5 

06.6 

16.5 

29.4 

45.7 

29 

30 

3503.3 

3598.4 

3695.6 

3795.1 

3896.9 

4001.3 

4108.4 

4218.3 

4331.3 

4447.6 

30 

31 

04.9 

3600.0 

97.3 

96.8 

98.6 

03.1 

10.2 

20.2 

33.2 

49.6 

31 

32 

06.5 

01.6 

98.9 

98.4 

3900.4 

04.8 

12.0 

22.0 

35.2 

51.6 

32 

33 

08.0 

03.2 

3700.5 

3800.1 

02.1 

06.6 

13.8 

23.9 

37.1 

53.5 

33 

34 

09.6 

04.8 

02.2 

01.8 

03.8 

08.3 

15.6 

25.8 

39.0 

55.5 

34 

35 

3511.2 

3606.4 

3703.8 

3803.5 

3905.5 

4010.1 

4117.4 

4227.6 

4340.9 

4457.5 

35 

36 

12.7 

08.0 

05.5 

05.1 

07.2 

11.9 

19.2 

29.5 

42.8 

59.4 

36 

37 

14.3 

09.6 

07.1 

06.8 

09.0 

13.6 

21.0 

31.3 

44.7 

61.4 

37 

38 

15.9 

11.2 

08.7 

08.5 

10.7 

15.4 

22.9 

33.2 

46.6 

63.4 

38 

39 

17.5 

12.8 

10.4 

10.2 

12.4 

17.2 

24.7 

35.1 

48.6 

65.4 

39 

40 

3519.0 

3614.5 

3712.0 

3811.9 

3914.1 

4018.9 

4126.5 

4236.9 

4350.5 

4467. 

40 

41 

20.6 

16.1 

13.7 

13.6 

15.9 

20.7 

28.3 

38.8 

52.4 

69. 

41 

42 

22.2 

17.7 

15.3 

15.2 

17.6 

22.5 

30.1 

40.7 

54.3 

71. 

42 

43 

23.7 

19.3 

17.0 

17.0 

19.3 

24.3 

31.9 

42.5 

56.2 

73. 

43 

44 

25.3 

20.9 

18.6 

18.6 

21.0 

26.0 

33.8 

44.4 

58.2 

75. 

44 

45 

3526.9 

3622.5 

3720.3 

3820.3 

3922.8 

4027.8 

4135.6 

4246.3 

4360.1 

4477. 

45 

46 

28.5 

24.1 

21.9 

22.0 

24.5 

29.6 

37.4 

48.1 

62.0 

79. 

46 

47 

30.1 

25.7 

23.6 

23.7 

26.2 

31.4 

39.2 

50.0 

63.9 

81. 

47 

48 

31.6 

27.3 

25.2 

25.4 

28.0 

33.1 

41.0 

51.9 

65.9 

83. 

48 

49 

33.2 

29.0 

26.9 

27.1 

29.7 

34.9 

42.9 

53.8 

67.8 

85. 

49 

50 

3534.8 

3630.6 

3728.5 

3828.7 

3931.4 

4036.7 

4144.7 

4255.6 

4369.7 

4487. 

50 

51 

36.4 

32.2 

30.2 

30.4 

33.2 

38.5 

46.5 

57.5 

71.7 

89. 

51 

52 

37.9 

33.8 

31.8 

32.1 

34.9 

40.2 

48.3 

59.4 

73.6 

91. 

52 

53 

39.5 

35.4 

33.5 

33.8 

36.6 

42.0 

50.2 

61.3 

75.5 

93. 

53 

54 

41.1 

37.0 

35.1 

35.5 

38.4 

43.8 

52.0 

63.1 

77.4 

95. 

54 

55 

3542.7 

3638.6 

3736.8 

3837.2 

3940.1 

4045.6 

4153.8 

4265.0 

4379.4 

4497. 

55 

56 

44.3 

40.3 

38.4 

38.9 

41.8 

47.4 

55.7 

66.9 

81.3 

99. 

56 

57 

45.9 

41.9 

40.1 

40.6 

43.6 

49.1 

57.5 

68.8 

83.2 

4501.1 

57 

58 

47.4 

43.5 

41.7 

42.3 

45.3 

50.9 

59.3 

70.7 

85.2 

03.1 

58 

59 

49.0 

45.1 

43.4 

45.0 

47.0 

52.7 

61.1 

72.5 

87.1 

05.1 

59 

60 

3550.6 

3646.7 

3745.1 

3845.7 

3948.8 

4054.5 

4163.0 

4274.4 

4389.1 

4507.1 

60 

' 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

/ 

Table  6 


Table  7       247 


Combined  Correction  for  Observed 
Sextant  Altitudes 


Correction  for  Dip  of 

Sea  Horizon 

(Sun  or  Star) 


OBSERVED 
ALTITUDE 

CORRECTION 

For  Sun   (to 
be  added  to 
observed  alti- 
tude) 

For  Star  (to 
be  subtracted 

from  observed 
altitude) 

5° 

6'  14" 

9'  55" 

6 

7  41 

8  28 

7 

8  45 

7  24 

8 

9  35 

6  34 

9 

10   16 

5  53 

10 

10  50 

5  19 

11 

11   17 

4  51 

12 

11  41 

4  27 

13 

12     2 

4     7 

14 

12   19 

3  49 

15 

12  34 

3  34 

20 

13  29 

2  39 

25 

14     3 

2     5 

30 

14  26 

1  41 

35 

14  44 

1   23 

40 

14  57 

1    10 

45 

15     8 

0  58 

50 

15   17 

0  49 

55 

15  25 

0  40 

60 

15  31 

0  34 

65 

15  37 

0  27 

70 

15  42 

0  21 

75 

15  47 

0  16 

80 

15  52 

0  10 

85 

15  55 

0     5 

HEIGHT  op 
OBSERVER'S 
EYE   ABOVE 
SEA  LEVEL 
(feet) 

DIP  CORREC- 
TION (to  be 
subtracted 

from 
observed 
altitude) 

4 

1'  58" 

6 

2  24 

8 

2  46 

10 

3  06 

12 

3  24 

14 

3  40 

16 

3  55 

18 

4     9 

20 

4  23 

22 

4  36 

24 

4  48 

26 

5     0 

28 

5  11 

30 

5  22 

35 

5  48 

40 

6   12 

45 

6  36 

50 

6  56 

55 

7  16 

60 

7  35 

70 

8  12 

85 

9     2 

100 

9  48 

Small  supplementary  correction,  for  Sun 
only. 

Jan.  to  March  \  ,  ,  in,, 
and  Oct.  toDec./addl°  • 
April  to  Sept.,  subtract  10". 


The  dip  correction  is  not 
required  when  the  artificial 
horizon  is  used. 


248 


Table  8 


To  Change  Hours  and  Minutes  into  Decimals  of  a  Day 


HOURS  EXPRESSED 

AS  DECIMAL  PARTS 

OF  A  DAY 


HOURS 

DECIMAL 

1 

.0416 

2 

.0833 

3 

.1250 

4 

.1666 

5 

.2083 

6 

.2500 

7 

.2916 

8 

.3333 

9 

.3750 

10 

.4166 

11 

.4583 

12 

.5000 

13 

.5416 

14 

.5833 

15 

.6249 

16 

.6666 

17 

.7083 

18 

.7500 

19 

.7916 

20 

.8333 

21 

.8749 

22 

.9166 

23 

.9583 

24 

1.0000 

MINUTES  EXPRESSED  AS  DECIMAL  PARTS 
OF  A  DAY 


MINUTES 

DECIMAL 

MINUTES 

DECIMAL 

1 

.0006 

31 

.0215 

2 

.0013 

32 

.0222 

3 

.0020 

33 

.0229 

4 

.0027 

34 

.0236 

5 

.0034 

35 

.0243 

6 

.0041 

36 

.0250 

7 

.0048 

37 

.0256 

8 

.0055 

38 

.0263 

9 

.0062 

39 

.0270 

10 

.0069 

40 

.0277 

11 

.0076 

41 

.0284 

12 

.0083 

42 

.0291 

13 

.0090 

43 

.0298 

14 

.0097 

44 

.0305 

15 

.0104 

45 

.0312 

16 

.0111 

46 

.0319 

17 

.0118 

47 

.0326 

18 

.0125 

48 

.0333 

19 

.0131 

49 

.0340 

20 

.0138 

50 

.0347 

21 

.0145 

51 

.0354 

22 

.0152 

52 

.0361 

23 

.0159 

53 

.0368 

24 

.0166 

54 

.0375 

25 

.0173 

55 

.0381 

26 

.0180 

56 

.0388 

27 

.0187 

57 

.0395 

28 

.0194 

58 

.0402 

29 

.0201 

59 

.0409 

30 

.0208 

60 

.0416 

Table  9 


249 


To  Interchange  Degrees  and  Minutes  of  Longitude  and  Hours,  Minutes, 
and  Seconds  of  Time.    Part  1 


0 

1* 

2* 

3* 

4h 

6* 

6* 

7* 

8* 

9* 

10* 

11* 

Qm 

0° 

15° 

30° 

45° 

60° 

75° 

90° 

105° 

120° 

135° 

150° 

165° 

4 

1 

16 

31 

46 

61 

76 

91 

106 

121 

136 

151 

166 

8 

2 

17 

32 

47 

62 

77 

92 

107 

122 

137 

152 

167 

12 

3 

18 

33 

48 

63 

78 

93 

108 

123 

138 

153 

168 

16 

4 

19 

34 

49 

64 

79 

94 

109 

124 

139 

154 

169 

20 

5 

20 

35 

50 

65 

80 

95 

110 

125 

140 

155 

170 

24 

6 

21 

36 

51 

66 

81 

96 

111 

126 

141 

156 

171 

28 

7 

22 

37 

52 

67 

82 

97 

112 

127 

142 

157 

172 

32 

8 

23 

38 

53 

68 

83 

98 

113 

128 

143 

158 

173 

36 

9 

24 

39 

54 

69 

84 

99 

114 

129 

144 

159 

174 

40 

10 

25 

40 

55 

70 

85 

100 

115 

130 

145 

160 

175 

44 

11 

26 

41 

56 

71 

86 

101 

116 

131 

146 

161 

176 

48 

12 

27 

42 

57 

72 

87 

102 

117 

132 

147 

162 

177 

52 

13 

28 

43 

58 

73 

88 

103 

118 

133 

148 

163 

178 

:><; 

14 

29 

44 

59 

74 

89 

104 

119 

134 

149 

164 

179 

12* 

13* 

I4h 

15* 

16* 

17* 

18* 

19* 

20* 

21* 

22* 

23* 

Om 

180° 

195° 

210° 

225° 

240° 

255° 

270° 

285° 

300° 

315° 

330° 

345° 

4 

181 

196 

211 

226 

241 

256 

271 

286 

301 

316 

331 

346 

8 

182 

197 

212 

227 

242 

257 

272 

287 

302 

317 

332 

347 

12 

183 

198 

213 

228 

243 

258 

273 

288 

303 

318 

333 

348 

16 

184 

199 

214 

229 

244 

259 

274 

289 

304 

319 

334 

349 

20 

185 

200 

215 

230 

245 

260 

275 

290 

305 

320 

335 

350 

24 

186 

201 

216 

231 

246 

261 

276 

291 

306 

321 

336 

351 

28 

187 

202 

217 

232 

247 

262 

277 

292 

307 

322 

337 

352 

32 

188 

203 

218 

233 

248 

263 

278 

293 

308 

323 

338 

353 

36 

189 

204 

219 

234 

249 

264 

279 

294 

309 

324 

339 

354 

40 

190 

205 

220 

235 

250 

265 

280 

295 

310 

325 

340 

355 

44 

191 

206 

221 

236 

251 

266 

281 

296 

311 

326 

341 

356 

48 

192 

207 

222 

237 

252 

267 

282 

297 

312 

327 

342 

357 

52 

193 

208 

223 

238 

253 

268 

283 

298 

313 

328 

343 

358 

56 

194 

209 

224 

239 

254 

269 

284 

299 

314 

329 

344 

359 

Part  2 


EXPLANATION  OF  TABLE  9 

1.  To   change   degrees   of   longitude   into   hours   and 
minutes  of  time:    Find  the  number  of  degrees  in  Part  1. 
The  required  hours  will  then  be  found  at  the  head  of  the 
column  containing  the  degrees,    and  the   required  min- 
utes at  the   left-hand   end   of  the   line   containing   the 
degrees. 

Examples:    113°  =  7*  32m ;    294°  =  19*36™. 

2.  To  change  minutes  of  longitude  into  minutes  and 
seconds  of  time  :  Find  the  minutes  of  longitude  in  Part  2. 
The  required  minutes  and  seconds  of  time  will  again 
be  found  at  the  head  of  the  column  and  the  left-hand  end 
of  the  line. 

Examples :  43'  =  2m  52'  •   28'  =  lm  52*. 

3.  1  and  2  can  be  combined  by  addition. 

Examples  :    113°  43'  =  lh  34m  52*. 
294°  28'  =  19h  37m  52*. 

4.  To  change  hours  and  minutes  of  time  into  degrees 
and  minutes  of  longitude :    Find  the  number  of  hours  at 
the  head  of  one  of  the  columns  of  Part  1 ;  then  run  down 
the  column  until  you  reach  a  line  having  at  its  left-hand 
end  a  number  of  minutes  equal  to  (or  just  smaller  than) 
the  given  number  of  minutes  of  time.    Where  that  line 

and  column  meet  you  will  find  the  required  degrees  of  longitude. 

Examples:   lh  32m  =  113°;    19*  36m  =294°. 

5.  To  change  minutes  and  seconds  of  time  into  minutes  of  longitude  :   Find  the  number  of 
minutes  of  time  at  the  head  of  one  of  the  columns  of  Part  2  ;  then  run  down  the  column  until 
you  reach  a  line  having  at  its  left-hand  end  a  number  of  seconds  equal  (or  nearly  equal)  to 
the  given  number  of  seconds  of  time.     Where  that  line  and  column  meet  you  will  find  the 
minutes  of  longitude. 

Examples :   2m  52*  =  43' ;    lm  52'  =  28'. 

6.  4  and  5  can  be  combined  by  addition : 

Examples:  7»34m52'  =  113° 43';  19/'37m52«  =294° 28'. 


Q,n 

lm 

2m 

3m 

0« 

0' 

15' 

30' 

45' 

4 

1 

16 

31 

46 

8 

2 

17 

32 

47 

12 

3 

18 

33 

48 

16 

4 

19 

34 

49 

20 

5 

20 

35 

50 

24 

6 

21 

36 

51 

28 

7 

22 

37 

52 

32 

8 

23 

38 

53 

36 

9 

24 

39 

54 

40 

10 

25 

40 

55 

44 

11 

26 

41 

56 

48 

12 

27 

42 

57 

52 

13 

28 

43 

58 

56 

14 

29 

44 

59 

250 


Table  10.    Haversine  Table 


s    ' 

0*  Om     0° 

Oh  4m.       10 

Oh  sm    2° 

Oh  jgm     30 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hay. 

No. 

0   0 

0.00000 

5.88168 

0.00008 

6.48371 

0.00030 

6.83584 

0.00069 

4   1 

2.32539 

.00000 

.89604 

.00008 

.49092 

.00031 

.84065 

.00069 

8   2 

.92745 

.00000 

.91016 

.00008 

.49807 

.00031 

.84543 

.00070 

12   3 

3.27963 

.00000 

.92406 

.00008 

.50516 

.00032 

.85019 

.00071 

16   4 

.52951 

.00000 

.93774 

.00009 

.51219 

.00033 

.85492 

.00072 

20   5 

3.72333 

0.00000 

5.95121 

0.00009 

6.51916 

0.00033 

6.85963 

0.00072 

24   6 

.88169 

.00000 

.96447 

.00009 

.52608 

.00034 

.86431 

.00073 

28   7 

4.01559 

.00000 

.97753 

.00010 

.53295 

.00034 

.86897 

.00074 

32   8 

.13157 

.00000 

.99040 

.00010 

.53976 

.00035 

.87360 

.00075 

36   9 

.23388 

.00000 

6.00308 

.00010 

.54652 

.00035 

.87821 

.00076 

40  10 

4.32539 

0.00000 

6.01557 

0.00010 

6.55323 

0.00036 

6.88279 

0.00076 

44  11 

.40818 

.00000 

.02789 

.00011 

.55988 

.00036 

.88735 

.00077 

48  12 

.48375 

.00000 

.04004 

.00011 

.56649 

.00037 

.89188 

.00078 

52  13 

.55328 

.00000 

.05202 

.00011 

.57304 

.00037 

.89639 

.00079 

56  14 

.61765 

.00000 

.06384 

.00012 

.57955 

.00038 

.90088 

.00080 

s    ' 

Oh  lm     0° 

Oh  5m     1° 

Oh  9m      2° 

Qh  I3m      30 

0  15 

4.67757 

0.00000 

6.07550 

0.00012 

6.58600 

0.00039 

6.90535 

0.00080 

4  16 

.73363 

.00001 

.08700 

.00012 

.59241 

.00039 

.90979 

.00081 

5  17 

.78629 

.00001 

.09836 

.00013 

.59878 

.00040 

.91421 

.00082 

12  18 

.83594 

.00001 

.10956 

.00013 

.60509 

.00040 

.91860 

.00083 

^S  19 

.88290 

.00001 

.12063 

.00013 

.61136 

.00041 

.92298 

.00084 

20  20 

4.92745 

0.00001 

6.13155 

0.00014 

6.61759 

0.00041 

6.92733 

0.00085 

24  21 

.96983 

.00001 

.14234 

.00014 

.62377 

.00042 

.93166 

.00085 

25  22 

5.01024 

.00001 

.15300 

.00014 

.62991 

.00043 

.93597 

.00086 

32  23 

.04885 

.00001 

.16353 

.00015 

.63600 

.00043 

.94026 

.00087 

3£  24 

.08581 

.00001 

.17393 

.00015 

.64205 

.00044 

.94453 

.00088 

4(9  25 

5.12127 

0.00001 

6.18421 

0.00015 

6.64806 

0.00044 

6.94877 

0.00089 

44  26 

.15534 

.00001 

.19437 

.00016 

.65403 

.00045 

.95300 

.00090 

45  27 

.18812 

.00002 

.20441 

.00016 

.65996 

.00046 

.95720 

.00091 

52  28 

.21971 

.00002 

.21433 

.00016 

.66585 

.00046 

.96139 

.00091 

5£  29 

.25019 

.00002 

.22415 

.00017 

.67170 

.00047 

.96555 

.00092 

s    ' 

Qhgm      QO 

Qh  Qm      jo 

QK  iom    2° 

Qh  Ijrm      30 

0  30 

5.27963 

0.00002 

6.23385 

0.00017 

6.67751 

0.00048 

6.96970 

0.00093 

4  31 

.30811 

.00002 

.24345 

.00018 

.68328 

.00048 

.97382 

.00094 

8  32 

.33569 

.00002 

.25294 

.00018 

.68901 

.00049 

.97793 

.00095 

^2  33 

.36242 

.00002 

.26233 

.00018 

.69470 

.00050 

.98201 

.00096 

.70  34 

.38835 

.00002 

.27162 

.00019 

.70036 

.00050 

.98608 

.00097 

20  35 

5.41352 

0.00003 

6.28081 

0.00019 

6.70598 

0.00051 

6.99013 

0.00098 

24  36 

.43799 

.00003 

.28991 

.00019 

.71157 

.00051 

.99416 

.00099 

28  37 

.46179 

.00003 

.29891 

.00020 

.71712 

.00052 

.99817 

.00100 

32  38 

.48496 

.00003 

.30781 

.00020 

.72263 

.00053 

7.00216 

.00101 

35  39 

.50752 

.00003 

.31663 

.00021 

.72811 

.00053 

.00613 

.00101 

40  40 

5.52951 

0.00003 

6.32536 

0.00021 

6.73355 

0.00054 

7.01009 

0.00102 

44  41 

.55095 

.00004 

.33400 

.00022 

.73896 

.00055 

.01403 

.00103 

48  42 

.57189 

.00004 

.34256 

.00022 

.74434 

.00056 

.01795 

.00104 

52  43 

.59232 

.00004 

.35103 

.00022 

.74969 

.00056 

.02185 

.00105 

5£  44 

.61229 

.00004 

.35943 

.00023 

.75500 

.00057 

.02573 

.00106 

s 

Qh  3m       QO 

Qh  7m       10 

Qh  Hm     2° 

0^  15m    3° 

0  45 

5.63181 

0.00004 

6.36774 

0.00023 

6.76028 

0.00058 

7.02960 

0.00107 

4  46 

.65090 

.00004 

.37597 

.00024 

.76552 

.00058 

.03345 

.00108 

8  47 

.66958 

.00005 

.38412 

.00024 

.77074 

.00059 

.03729 

.00109 

.72  48 

.68787 

.00005 

.39220 

.00025 

.77592 

.00060 

.04110 

.00110 

16  49 

.70578 

.00005 

.40021 

.00025 

.78108 

.00060 

.04490 

.00111 

20  50 

5.72332 

0.00005 

6.40814 

0.00026 

6.78620 

0.00061 

7.04869 

0.00112 

24  51 

.74052 

.00006 

.41600 

.00026 

.79129 

.00062 

.05245 

.00113 

28  52 

.75739 

.00006 

.42379 

.00027 

.79630 

.00063 

.05620 

.00114 

32  53 

.77394 

.00006 

.43151 

.00027 

.80139 

.00063 

.05994 

.00115 

36  54 

.79017 

.00006 

.43916 

.00027 

.80640 

.00064 

.06366 

.00116 

40  55 

5.80611 

0.00006 

6.44675 

0.00028 

6.81137 

0.00065 

7.06736 

0.00117 

44  56 

.82176 

.00007 

.45427  • 

.00028 

.81632 

.00066 

.07105 

.00118 

48  57 

.83713 

.00007 

.46172 

.00029 

.82124 

.00066 

.07472 

.00119 

52  58 

.85224 

.00007 

.46911 

.00029 

.82614 

.00067 

.07837 

.00120 

56  59 

.86709 

.00007 

.47644 

.00030 

.83100 

.00068 

.08201 

.00121 

£0  60 

5.88168 

0.00008 

6.48371 

0.00030 

6.83584 

0.00069 

7.08564 

0.00122 

Table  10.    Haversine  Table 


251 


s    / 

Oh  iem    4° 

Oh  20m    5° 

Oh  24m    6° 

Qh  28m      7° 

Hay. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

7.08564 

0.00122 

7.27936 

0.00190 

7.43760 

0.00274 

7.57135 

0.00373 

4   1 

.08925 

.00123 

.28225 

.00192 

.44001 

.00275 

.57341 

.00374 

8   2 

.09284 

.00124 

.28513 

.00193 

.44241 

.00277 

.57547 

.00376 

t*   3 

.09642 

.00125 

.28800 

.00194 

.44480 

.00278 

.57752 

.00378 

16   4 

.09999 

.00126 

.29086 

.00195 

.44719 

.00280 

.57957 

.00380 

20   5 

7.10354 

0.00127 

7.29371 

0.00197 

7.44957 

0.00282 

7.58162 

0.00382 

24   6 

.10708 

.00128 

.29655 

.00198 

.45194 

.00283 

.58366 

.00383 

28   7 

.11060 

.00129 

.29938 

.00199 

.45431 

.00285 

.58569 

.00385 

32   8 

.11411 

.00130 

.30220 

.00201 

.45667 

.00286 

.58772 

.00387 

36   9 

.11760 

.00131 

.30502 

.00202 

.45903 

.00288 

.58974 

.00389 

40  10 

7.12108 

0.00132 

7.30782 

0.00203 

7.46138 

0.00289 

7.59176 

0.00391 

44  11 

.12455 

.00133 

.31062 

.00204 

.46372 

.00291 

.59378 

.00392 

48  12 

.12800 

.00134 

.31340 

.00206 

.46605 

.00292 

.59579 

.00394 

52  13 

.13144 

.00135 

.31618 

.00207 

.46838 

.00294 

.59779 

.00396 

56  14 

.13486 

.00136 

.31895 

.00208 

.47071 

.00296 

.59979 

.00398 

s   ' 

Qh  1?™       40 

Qh  21m    5° 

Oh  25m    6° 

0*29m    7° 

0  15 

7.13827 

0.00137 

7.32171 

0.00210 

7.47302 

0.00297 

7.60179 

0.00400 

4  16 

.14167 

.00139 

.32446 

.00211 

.47533 

.00299 

.60378 

.00402 

5  17 

.14506 

.00140 

.32720 

.00212 

.47764 

.00300 

.60577 

.00403 

12  18 

.14843 

.00141 

.32994 

.00214 

.47994 

.00302 

.60775 

.00405 

^5  19 

.15179 

.00142 

.33266 

.00215 

.48223 

.00304 

.60973 

.00407 

20  20 

7.15513 

0.00143 

7.33538 

0.00216 

7.48452 

0.00305 

7.61170 

0.00409 

24  21 

.15846 

.00144 

.33809 

.00218 

.48680 

.00307 

.61367 

.00411 

25  22 

.16178 

.00145 

.34079 

.00219 

.48907 

.00308 

.61564 

.00413 

32  23 

.16509 

.00146 

.34348 

.00221 

.49134 

.00310 

.61760 

.00415 

35  24 

.16839 

.00147 

.34616 

.00222 

.49360 

.00312 

.61955 

.00416 

40  25 

7.17167 

0.00148 

7.34884 

0.00223 

7.49586 

0.00313 

7.62151 

0.00418 

44  26 

.17494 

.00150 

.35150 

.00225 

.49811 

.00315 

.62345 

.00420 

4<S  27 

.17820 

.00151 

.35416 

.00226 

.50036 

.00316 

.62540 

.00422 

52  28 

.18144 

.00152 

.35681 

.00227 

.50259 

.00318 

.62733 

.00424 

,56  29 

.18468 

.00153 

.35945 

.00229 

.50483 

.00320 

.62927 

.00426 

s 

4° 

Qh  22m      5° 

Qh  2&n      6° 

Oh  som    7° 

0  30 

7.18790 

0.00154 

7.36209 

0.00230 

7.50706 

0.00321 

7.63120 

0.00428 

4  31 

.19111 

.00155 

.36471 

.00232 

.50928 

.00323 

.63312 

.00430 

5  32 

.19430 

.00156 

.36733 

.00233 

.51149 

.00325 

.63504 

.00432 

12  33 

.19749 

.00158 

.36994 

.00234 

.51370 

.00326 

.63696 

.00433 

J5  34   .20066 

.00159 

.37254 

.00236 

.51591 

.00328 

.63887 

.00435 

20  35 

7.20383 

0.00160 

7.37514 

0.00237 

7.51811 

0.00330 

7.64078 

0.00437 

24  36 

.20698 

.00161 

.37773 

.00239 

.52030 

.00331 

.64269 

.00439 

28  37 

.21012 

.00162 

.38030 

.00240 

.52249 

.00333 

.64458 

.00441 

32  38 

.21325 

.00163 

.38288 

.00241 

.52467 

.00335 

.64648 

.00443 

35  39 

.21636 

.00165 

.38544 

.00243 

.52685 

.00336 

.64837 

.00445 

40  40 

7.21947 

0.00166 

7.38800 

0.00244 

7.52902 

0.00338 

7.65026 

0.00447 

44  41 

.22256 

.00167 

.39054 

.00246 

.53119 

.00340 

.65214 

.00449 

48  42 

.22565 

.00168 

.39309 

.00247 

.53335 

.00341 

.65402 

.00451 

.72  43 

.22872 

.00169 

.39562 

.00249 

.53550 

.00343 

.65590 

.00453 

55  44 

.23178 

.00171 

.39815 

.00250 

.53766 

.00345 

.65777 

.00455 

s   ' 

Oh  igm    4° 

5° 

Qh  27m      6° 

Qh  31m      7° 

0  45 

7.23483 

0.00172 

7.40067 

0.00252 

7.53980 

0.00347 

7.65964 

0.00457 

4  46 

.23787 

.00173 

.40318 

.00253 

.54194 

.00348 

.66150 

.00459 

5  47 

.24090 

.00174 

.40568 

.00255 

.54407 

.00350 

.66336 

.00461 

72  48 

.24392 

.00175 

.40818 

.00256 

.54620 

.00352 

.66521 

.00463 

16  49 

.24693 

.00177 

.41067 

.00257 

.54833 

.00353 

.66706 

.00465 

20  50 

7.24993 

0.00178 

7.41315 

0.00259 

7.55045 

0.00355 

7.66891 

0.00467 

24  51 

.25292 

.00179 

.41563 

.00260 

.55256 

.00357 

.67075 

.00469 

28  52 

.25590 

.00180 

.41810 

.00262 

.55467 

.00359 

.67259 

.00471 

32  53 

.25886 

.00181 

.42056 

.00263 

.55677 

.00360 

.67443 

.00473 

35  54 

.26182 

.00183 

.42301 

.00265 

.55887 

.00362 

.67626 

.00475 

40  55 

7.26477 

0.00184 

7.42546 

0.00266 

7.56096 

0.00364 

7.67809 

0.00477 

44  56 

.26771 

.00185 

.42790 

.00268 

.56305 

.00366 

.67991 

.00479 

48  57 

.27064 

.00186 

.43034 

.00269 

.56513 

.00367 

.68173 

.00481 

52  58 

.27355 

.00188 

.43277 

.00271 

.56721 

.00369 

.68355 

.00483 

55  59 

.27646 

.00189 

.43519 

.00272 

.56928 

.00371 

.68536 

.00485 

50  60 

7.27936 

0.00190 

7.43760 

0.00274 

7.57135 

0.00373 

7.68717 

0.00487 

252 


Table  10.    Haversine  Table 


s   ' 

Oh  32m    go 

Qh  Sffn      90 

0^  40m    10° 

0^  44m    11° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

7.68717 

0.00487 

7.78929 

0.00616 

7.88059 

0.00760 

7.96315 

0.00919 

4   1 

.68897 

.00489 

.79089 

.00618 

.88203 

.00762 

.96446 

.00921 

8   2 

.69077 

.00491 

.79249 

.00620 

.88348 

.00765 

.96577 

.00924 

12   3 

.69257 

.00493 

.79409 

.00622 

.88491 

.00767 

.96707 

.00927 

16   4 

.69437 

.00495 

.79568 

.00625 

.88635 

.00770 

.96838 

.00930 

20   5 

7.69616 

0.00497 

7.79728 

0.00627 

7.88778 

0.00772 

7.96968 

0.00933 

24   6 

.69794 

.00499 

.79886 

.00629 

.88921 

.00775 

.97098 

.00935 

28   7 

.69972 

.00501 

.80045 

.00632 

.89064 

.00777 

.97228 

.00938 

32   8 

.70150 

.00503 

.80203 

.00634 

.89207 

.00780 

.97358 

.00941 

36   9 

.70328 

.00505 

.80361 

.00636 

.89349 

.00783 

.97478 

.00944 

40  10 

7.70505 

0.00507 

7.80519 

0.00639 

7.89491 

0.00785 

7.97617 

0.00947 

44  11 

.70682 

.00509 

.80677 

.00641 

.89633 

.00788 

.97746 

.00949 

48  12 

.70858 

.00511 

.80834 

.00643 

.89775 

.00790 

.97875 

.00952 

,52  13 

.71034 

.00513 

.80991 

.00646 

.89916 

.00793 

.98003 

.00955 

55   14 

.71210 

.00515 

.81147 

.00648 

.90057 

.00795 

.98132 

.00958 

s    ' 

Oh  S3m    8° 

Qh  37m    9° 

Oh  4lm    10° 

Oh  45™    11° 

f  15 

7.71385 

0.00517 

7.81303 

0.00650 

7.90198 

0.00798 

7.98260 

0.00961 

4  16 

.71560 

.00520 

.81459 

.00653 

.90339 

.00801 

.98389 

.00964 

5  17 

.71735 

.00522 

.81615 

.00655 

.90480 

.00803 

.98517 

.00966 

12  18 

.71909 

.00524 

.81771 

.00657 

.90620 

.00806 

.98644 

.00969 

J0  19 

.72083 

.00526 

.81926 

.00660 

.90760 

.00808 

.98772 

.00972 

20  20 

7.72257 

0.00528 

7.82081 

0.00662 

7.90900 

0.00811 

7.98899 

0.00975 

24  21 

.72430 

.00530 

.82235 

.00664 

.91039 

.00814 

.99027 

.00978 

25  22 

.72603 

.00532 

.82390 

00667 

.91179 

.00816 

.99154 

.00981 

32  23 

.72775 

.00534 

.82544 

.00669 

.91318 

.00819 

.99281 

.00984 

55  24 

.72948 

.00536 

.82698 

.00671 

.91457 

.00821 

.99407 

.00986 

40  25 

7.73119 

0.00539 

7.82851 

0.00674 

7.91596 

0.00824 

7.99534 

0.00989 

44  26 

.73291 

.00541 

.83004 

.00676 

.91734 

.00827 

.99660 

.00992 

45  27 

.73462 

.00543 

.83157 

.00679 

.91872 

.00829 

.99786 

.00995 

52  28 

.73633 

.00545 

.83310 

.00681 

.92010 

.00832 

.99912 

.00998 

off  29 

.73803 

.00547 

.83463 

.00683 

.92148 

.00835 

8.00038 

.01001 

s    ' 

Oh  34™     8° 

Qh  38m      90 

o^  42™   10° 

Qh  46™      11° 

0  30 

7.73974 

0.00549 

7.83615 

0.00686 

7.92286 

0.00837 

8.00163 

0.01004 

4  31 

.74143 

.00551 

.83767 

.00688 

.92423 

.00840 

.00289 

.01007 

8  32 

.74313 

.00554 

.83918 

.00691 

.92560 

.00843 

.00414 

.01010 

^2  33 

.74482 

.00556 

.84070 

.00693 

.92697 

.00845 

.00539 

.01012 

J5  34 

.74651 

.00558 

.84221 

.00695 

.92834 

.00848 

.00664 

.01015 

20  35 

7.74819 

0.00560 

7.84372 

0.00698 

7.92970 

0.00851 

8.00788 

0.01018 

24  36 

.74988 

.00562 

.84522 

.00700 

.93107 

.00853 

.00913 

.01021 

28  37 

.75155 

.00564 

.84672 

.00703 

.93243 

.00856 

.01037 

.01024 

32  38 

.75323 

.00567 

.84822 

.00705 

.93379 

.00859 

.01161 

.01027 

3£  39 

.75490 

.00569 

.84972 

.00707 

.93514 

.00861 

.01285 

.01030 

40  40 

7.75657 

0.00571 

7.85122 

0.00710 

7.93650 

0.00864 

8.01409 

0.01033 

44  41 

.75824 

.00573 

.85271 

.00712 

.93785 

.00867 

.01532 

.01036 

48  42 

.75990 

.00575 

.85420 

.00715 

.93920 

.00869 

.01656 

.01039 

52  43 

.76156 

.00578 

.85569 

.00717 

.94055 

.00872 

.01779 

.01042 

55  44 

.76321 

.00580 

.85717 

.00720 

.94189 

.00875 

.01902 

.01045 

s    ' 

0*  35m     8° 

Qh  3gm    90 

Oh  43m    10° 

Oh  47m    11° 

0  45 

7.76487 

0.00582 

7.85866 

0.00722 

7.94324 

0.00877 

8.02025 

0.01048 

4  46 

.76652 

.00584 

.86014 

.00725 

.94458 

.00880 

.02148 

.01051 

S  47 

.76816 

.00586 

.86161 

.00727 

.94592 

.00883 

.02270 

.01054 

^2  48 

.76981 

.00589 

.86309 

.00730 

.94726 

.00886 

.02392 

.01057 

16  49 

.77145 

.00591 

.86456 

.00732 

.94859 

.00888 

.02515 

.01060 

20  50 

7.77308 

0.00593 

7.86603 

0.00735 

7.94992 

0.00891 

8.02637 

0.01063 

24  51 

.77472 

.00595 

.86750 

.00737 

.95126 

.00894 

.02758 

.01066 

25  52 

.77635 

.00598 

.86896 

.00740 

.95259 

.00897 

.02880 

.01069 

32  53 

.77798 

.00600 

.87042 

.00742 

.95391 

.00899 

.03001 

.01072 

36  54 

.77960 

.00602 

.87188 

.00745 

.95524 

.00902 

.03123 

.01075 

40  55 

7.78122 

0.00604 

7.87334 

0.00747 

7.95656 

0.00905 

8.03244 

0.01078 

44  56 

.78284 

.00607 

.87480 

.00750 

.95788 

.00908 

.03365 

.01081 

45  57 

.78446 

.00609 

.87625 

.00752 

.95920 

.00910 

.03486 

.01084 

52  58 

.78607 

.00611 

.87770 

.00755 

.96052 

.00913 

.03606 

.01087 

56  59 

.78768 

.00613 

.87915 

.00757 

.96183 

.00916 

.03727 

.01090 

£0  60 

7.78929 

0.00616 

7.88059 

0.00760 

7.96315 

0.00919 

8.03847 

0.01093 

Table  10.    Haversine  Table 


253 


s    ' 

Oh  45"1    12° 

Oh  52m    13° 

Qh  Q&n      14° 

lh  Q™      15° 

Ha?. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.03847 

0.01093 

8.10772 

0.01282 

8.17179 

0.01485 

8.23140 

0.01704 

4   1 

.03967 

.01096 

.10883 

.01285 

.17282 

.01489 

.23235 

.01707 

8   2 

.04087 

.01099 

.10993 

.01288 

.17384 

.01492 

.23331 

.01711 

12   3 

.04207 

.01102 

.11104 

.01291 

.17487 

.01496 

.23427 

.01715 

16   4 

.04326 

.01105 

.11214 

.01295 

.17590 

.01499 

.23523 

.01719 

20   5 

8.04446 

0.01108 

8.11324 

0.01298 

8.17692 

0.01503 

8.23618 

0.01723 

24   6 

.04565 

.01111 

.11435 

.01301 

.17794 

.01506 

.23713 

.01726 

28   7 

.04684 

.01114 

.11544 

.01305 

.17896 

.01510 

.23809 

.01730 

32   8 

.04803 

.01117 

.11654 

.01308 

.17998 

.01513 

.23904 

.01734 

36   9 

.04922 

.01120 

.11764 

.01311 

.18100 

.01517 

.23999 

.01738 

40  10 

8.05041 

0.01123 

8.11873 

0.01314 

8.18202 

0.01521 

8.24094 

0.01742 

44  11 

.05159 

.01126 

.11983 

.01317 

.18303 

.01524 

.24189 

.01745 

48  12 

.05277 

.01129 

.12092 

.01321 

.18405 

.01528 

.24283 

.01749 

.52  13 

.05395 

.01132 

.12201 

.01324 

.18506 

.01531 

.24378 

.01753 

56  14 

.05513 

.01135 

.12310 

.01328 

.18607 

.01535 

.24473 

.01757 

s    ' 

0*49™    12° 

0^  53™    13° 

Oh  5jm    14° 

lh  I*"      15° 

0  15 

8.05631 

0.01138 

8.12419 

0.01331 

8.18709 

0.01538 

8.24567 

0.01761 

4  16 

.05749 

.01142 

.12528 

.01334 

.18810 

.01542 

.24661 

.01764 

5  17 

.05866 

.01145 

.12636 

.01338 

.18910 

.01546 

.24755 

.01768 

12  18 

.05984 

.01148 

.12745 

.01341 

.19011 

.01549 

.24850 

.01772 

^6  19 

.06101 

.01151 

.12853 

.01344 

.19112 

.01553 

.24944 

.01776 

20  20 

8.06218 

0.01154 

8.12961 

0.01348 

8.19212 

0.01556 

8.25037 

0.01780 

24  21 

.06335 

.01157 

.13069 

.01351 

.19313 

.01560 

.25131 

.01784 

25  22 

.06451 

.01160 

.13177 

.01354 

.19413 

.01564 

.25225 

.01788 

32  23 

.06568 

.01163 

.13285 

.01358 

.19513 

.01567 

.25319 

.01791 

36  24 

.06684 

.01166 

.13392 

.01361 

.19613 

.01571 

.25412 

.01795 

40  25 

8.06800 

0.01170 

8.13500 

0.01365 

8.19713 

0.01574 

8.25505 

0.01799 

44  26 

.06917 

.01173 

.13607 

.01368 

.19813 

.01578 

.25599 

.01803 

45  27 

.07032 

.01176 

.13714 

.01371 

.19913 

.01582 

.25692 

.01807 

52  28 

.07148 

.01179 

.13822 

.01375 

.20012 

.01585 

.25785 

.01811 

56'  29 

.07264 

.01182 

.13928 

.01378 

.20112 

.01589 

.25878 

.01815 

s    ' 

Oh  50™    12° 

Oh  54m    13° 

Oh  58™    14° 

lh  21"    15° 

0  30 

8.07379 

0.01185 

8.14035 

0.01382 

8.20211 

0.01593 

8.25971 

0.01818 

4  31 

.07494 

.01188 

.14142 

.01385 

.20310 

.01596 

.26064 

.01822 

8  32 

.07610 

.01192 

.14248 

.01388 

.20410 

.01600 

.26156 

.01826 

J2  33 

.07725 

.01195 

.14355 

.01392 

.20509 

.01604 

.26249 

.01830 

J0  34 

.07839 

.01198 

.14461 

.01395 

.20608 

.01607 

.26341 

.01834 

20  35 

8.07954 

0.01201 

8.14567 

0.01399 

8.20706 

0.01611 

8.26434 

0.01838 

24  36 

.08069 

.01204 

.14673 

.01402 

.20805 

.01615 

.26526 

.01842 

28  37 

.08183 

.01207 

.14779 

.01405 

.20904 

.01618 

.26618 

.01846 

32  38 

.08297 

.01211 

.14885 

.01409 

.21002 

.01622 

.26710 

.01850 

36  39 

.08411 

.01214 

.14991 

.01412 

.21100 

.01626 

.26802 

.01854 

40  40 

8.08525 

0.01217 

8.15096 

0.01416 

8.21199 

0.01629 

8.26894 

0.01858 

44  41 

.08639 

.01220 

.15201 

.01419 

.21297 

.01633 

.26986 

.01861 

48  42 

.08752 

.01223 

.15307 

.01423 

.21395 

.01637 

.27078 

.01865 

52  43 

.08866 

.01226 

.15412 

.01426 

.21493 

.01640 

.27169 

.01869 

56  44 

.08979 

.01230 

.15517 

.01429 

.21590 

.01644 

.27261 

.01873 

s    ' 

12° 

13° 

14° 

lh  3m    15° 

0  45 

8.09092 

0.01233 

8.15622 

0.01433 

8.21688 

0.01648 

8.27352 

0.01877 

4  46 

.09205 

.01236 

.15726 

.01436 

.21785 

.01651 

.27443 

.01881 

<§  47 

.09318 

.01239 

.15831 

.01440 

.21883 

.01655 

.27534 

.01885 

J2  48 

.09431 

.01243 

.15935 

.01443 

.21980 

.01659 

.27626 

.01889 

16  49 

.09543 

.01246 

.16040 

.01447 

.22077 

.01663 

.27717 

.01893 

20  50 

8.09656 

0.01249 

8.16144 

0.01450 

8.22175 

0.01666 

8.27807 

0.01897 

24  51 

.09768 

.01252 

.16248 

.01454 

.22272 

.01670 

.27898 

.01901 

28  52 

.09880 

.01255 

.16352 

.01457 

.22368 

.01674 

.27989 

.01905 

32  53 

.09992 

.01259 

.16456 

.01461 

.22465 

.01677 

.28080 

.01909 

36  54 

.10104 

.01262 

.16559 

.01464 

.22562 

.01681 

.28170 

.01913 

40  55 

8.10216 

0.01265 

8.16663 

0.01468 

8.22658 

0.01685 

8.28260 

0.01917 

44  56 

.10327 

.01268 

.16766 

.01471 

.22755 

.01689 

.28351 

.01921 

48  57 

.10439 

.01272 

.16870 

.01475 

.22851 

.01692 

.28441 

.01925 

52  58 

.10550 

.01275 

.16973 

.01478 

.22947 

.01696 

.28531 

.01929 

56  59 

.10661 

.01278 

.17076 

.01482 

.23044 

.01700 

.28621 

.01933 

60  60 

8.10772 

0.01282 

8.17179 

0.01485 

8.23140 

0.01704 

8.28711 

0.01937 

254 


Table  10.    Haversine  Table 


s   ' 

JA  4m    16° 

lh  gm      17o 

lh  l^m      jgo 

lh  Iffn      19° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hay. 

No. 

0   0 

8.28711 

0.01937 

8.33940 

0.02185 

8.38867 

0.02447 

8.43522 

0.02724 

4   1 

.28801 

.01941 

.34025 

.02189 

.38946 

.02452 

.43597 

.02729 

8   2 

.28891 

.01945 

.34109 

.02193 

.39026 

.02456 

.43673 

.02734 

12   3 

.28980 

.01949 

.34194 

.02198 

.39105 

.02461 

.43748 

.02738 

16   4 

.29070 

.01953 

.34278 

.02202 

.39185 

.02465 

.43823 

.02743 

20   5 

8.29159 

0.01957 

8.34362 

0.02206 

8.39264 

0.02470 

8.43899 

0.02748 

24   6 

.29249 

.01961 

.34446 

.02210 

.39344 

.02474 

.43974 

.02753 

28   7 

.29338 

.01965 

.34530 

.02215 

.39423 

.02479 

.44049 

.02757 

32   8 

.29427 

.01969 

.34614 

.02219 

.39502 

.02483 

.44124 

.02762 

36   9 

.29516 

.01973 

.34698 

.02223 

.39581 

.02488 

.44199 

.02767 

40  10 

8.29605 

0.01977 

8.34782 

0.02227 

8.39660 

0.02492 

8.44273 

0.02772 

44  11 

.29694 

.01981 

.34865 

.02232 

.39739 

.02497 

.44348 

.02776 

48  12 

.29783 

.01985 

.34949 

.02236 

.39818 

.02501 

.44423 

.02781 

52  13 

.29872 

.01989 

.35032 

.02240 

.39897 

.02506 

.44498 

.02786 

56  14 

.29960 

.01993 

.35116 

.02245 

.39976 

.02510 

.44572 

.02791 

s 

lh  Qm    16° 

lh  gm    17° 

lh  igm    igo 

lh  ijm    19° 

0  15 

8.30049 

0.01998 

8.35199 

0.02249 

8.40055 

0.02515 

8.44647 

0.02796 

4  16 

.30137 

.02002 

.35282 

.02253 

.40133 

.02520 

.44721 

.02800 

5  17 

.30226 

.02006 

.35365 

.02258 

.40212 

.02524 

.44796 

.02805 

.72  18 

.30314 

.02010 

.35449 

.02262 

.40290 

.02529 

.44870 

.02810 

16  19 

.30402 

.02014 

.35532 

.02266 

.40369 

.02533 

.44944 

.02815 

20  20 

8.30490 

0.02018 

8.35614 

0.02271 

8.40447 

0.02538 

8.45018 

0.02820 

24  21 

.30578 

.02022 

.35697 

.02275 

.40525 

.02542 

.45093 

.02824 

25  22 

.30666 

.02026 

.35780 

.02279 

.40603 

.02547 

.45167 

.02829 

52  23 

.30754 

.02030 

.35863 

.02284 

.40681 

.02552 

.45241 

.02834 

36  24 

.30842 

.02034 

.35945 

.02288 

.40760 

.02556 

.45315 

.02839 

40  25 

8.30929 

0.02038 

8.36028 

0.02292 

8.40837 

0.02561 

8.45388 

0.02844 

44  26 

.31017 

.02043 

.36110 

.02297 

.40915 

.02565 

.45462 

.02849 

48  27 

.31104 

.02047 

.36193 

.02301 

.40993 

.02570 

.45536 

.02853 

52  28 

.31192 

.02051 

.36275 

.02305 

.41071 

.02575 

.45610 

.02858 

56  29 

.31279 

.02055 

.36357 

.02310 

.41149 

.02579 

.45683 

.02863 

s    ' 

lh  ffn     16° 

lh  IQ™    17° 

lh  14™    18° 

lh  is"1    19° 

0  30 

8.31366 

0.02059 

8.36439 

0.02314 

8.41226 

0.02584 

8.45757 

0.02868 

4  31 

.31453 

.02063 

.36521 

.02319 

.41304 

.02588 

.45830 

.02873 

5  32 

.31540 

.02067 

.36603 

.02323 

.41381 

.02593 

.45904 

.02878 

12  33 

.31627 

.02071 

.36685 

.02327 

.41459 

.02598 

.45977 

.02883 

1£  34 

.31714 

.02076 

.36767 

.02332 

.41536 

.02602 

.46050 

.02887 

20  35 

8.31800 

0.02080 

8.36849 

0.02336 

8.41613 

0.02607 

8.46124 

0.02892 

24  36 

.31887 

.02084 

.36930 

.02340 

.41690 

.02612 

.46197 

.02897 

25  37 

.31974 

.02088 

.37012 

.02345 

.41767 

.02616 

.46270 

.02902 

32  38 

.32060 

.02092 

.37093 

.02349 

.41845 

.02621 

.46343 

.02907 

35  39 

.32147 

.02096 

.37175 

.02354 

.41921 

.02626 

.46416 

.02912 

40  40 

8.32233 

0.02101 

8.37256 

0.02358 

8.41998 

0.02630 

8.46489 

0.02917 

44  41 

.32319 

.02105 

.37337 

.02363 

.42075 

.02635 

.46562 

.  02922 

4?  42 

.32405 

.02109 

.37419 

.02367 

.42152 

.02639 

.46634 

.02926 

52  43 

.32491 

.02113 

.37500 

.02371 

.42229 

.02644 

.46707 

.02931 

56  44 

.32577 

.02117 

.37581 

.02376 

.42305 

.02649 

.46780 

.02936 

s    ' 

lh  jm      16° 

lh  nm      17° 

lh  I5m      18° 

lh  I9m    19° 

0  45 

8.32663 

0.02121 

8.37662 

0.02380 

8.42382 

0.02653 

8.46852 

0.02941 

4  46 

.32749 

.02126 

.37742 

.02385 

.42458 

.02658 

.46925 

.02946 

8  47 

.32834 

.02130 

.37823 

.02389 

.42535 

.02663 

.46998 

.02951 

J2  48 

.32920 

.02134 

.37904 

.02394 

.42611 

.02668 

.47070 

.02956 

16  49 

.33006 

.02138 

.37985 

.02398 

.42687 

.02672 

.47142 

.02961 

20  50 

8.33091 

0.02142 

8.38065 

0.02402 

8.42764 

0.02677 

8.47215 

0.02966 

24  51 

.33176 

.02147 

.38146 

.02407 

.42840 

.02682 

.47287 

.02971 

28  52 

.33262 

.02151 

.38226 

.02411 

.42916 

.02686 

.47359 

.02976 

32  53 

.33347 

.02155 

.38306 

.02416 

.42992 

.02691 

.47431 

.02981 

36  54 

.33432 

.02159 

.38387 

.02420 

.43068 

.02696 

.47503 

.02986 

40  55 

8.33517 

0.02164 

8.38467 

0.02425 

8.43144 

0.02700 

8.47575 

0.02991 

44  56 

.33602 

.02168 

.38547 

.02429 

.43219 

.02705 

.47647 

.02996 

48  57 

.33686 

.02172 

.38627 

.02434 

.43295 

.02710 

.47719 

.03000 

52  58 

.33771 

.02176 

.38707 

.02438 

.43371 

.02715 

.47791 

.03005 

56  59 

.33856 

.02181 

.38787 

.02443 

.43446 

.02719 

.47862 

.03010 

60  60 

8.33940 

0.02185 

8.;:;^867 

0.02447 

8.43522 

0.02724 

8.47934 

0.03015 

Table  10.    Haversine  Table 


255 


5     ' 

jh  20m    20° 

lh  24m    21° 

jh  28m    22° 

jh  S2m    23° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.47934 

0.03015 

8.52127 

0.03321 

8.56120 

0.03641 

8.59931 

0.03975 

4   1 

.48006 

.03020 

.52195 

.03326 

.56185 

.03646 

.59993 

.03980 

8   2 

.48077 

.03025 

.52263 

.03331 

.56250 

.03652 

.60055 

.03986 

12   3 

.48149 

.03030 

.52331 

.03337 

.56315 

.03657 

.60117 

.03992 

16   4 

.48220 

.03035 

.52399 

.03342 

.56379 

.03663 

.60179 

.03998 

20   5 

8.48292 

0.03040 

8.52467 

0.03347 

8.56444 

0.03668 

8.60241 

0.04003 

24   6 

.48363 

.03045 

.52535 

.03352 

.56509 

.03674 

.60303 

.04009 

28   7 

.48434 

.03050 

.52602 

.03358 

.56574 

.03679 

.60365 

.04015 

32   8 

.48505 

.03055 

.52670 

.03363 

.56638 

.03685 

.60426 

.04020 

36   9 

.48576 

.03060 

.52738 

.03368 

.56703 

.03690 

.60488 

.04026 

40  10 

8.48648 

0.03065 

8.52806 

0.03373 

8.56767 

0.03695 

8.60550 

0.04032 

44  11 

.48719 

.03070 

.52873 

.03379 

.56832 

.03701 

.60611 

.04038 

48  12 

.48789 

.03075 

.52941 

.03384 

.56896 

.03706 

.60673 

.04043 

52  13 

.48860 

.03080 

.53008 

.03389 

.56960 

.03712 

.60734 

.04049 

56  14 

.48931 

.03085 

.53076 

.03394 

.57025 

.03717 

.60796 

.04055 

s    ' 

lh  21m    20° 

lh  25m    21° 

lh  29m    22° 

ih  33™    23° 

0  15 

8.49002 

0.03090 

8.53143 

0.03400 

8.57089 

0.03723 

8.60857 

0.04060 

4  16 

.49073 

.03095 

.53210 

.03405 

.57153 

.03728 

.60919 

.04066 

S  17 

.49143 

.03101 

.53277 

.03410 

.57217 

.03734 

.60980 

.04072 

12  18 

.49214 

.03106 

.53345 

.03415 

.57282 

.03740 

.61041 

.04078 

.70  19 

.49284 

.03111 

.53412 

.03421 

.57346 

.03745 

.61103 

.04083 

20  20 

8.49355 

0.03116 

8.53479 

0.03426 

8.57410 

0.03751 

8.61164 

0.04089 

24  21 

.49425 

.03121 

.53546 

.03431 

.57474 

.03756 

.61225 

.04095 

25  22 

.49496 

.03126 

.53613 

.03437 

.57538 

.03762 

.61286 

.04101 

32  23 

.49566 

.03131 

.53680 

.03442 

.57601 

.03767 

.61347 

.04106 

30  24 

.49636 

.03136 

.53747 

.03447 

.57665 

.03773 

.61408 

.04112 

40  25 

8.49706 

0.03141 

8.53814 

0.03453 

8.57729 

0.03778 

8.61469 

0.04118 

44  26 

.49777 

.03146 

.53880 

.03458 

.57793 

.03784 

.61530 

.04124 

48  27 

.49847 

.03151 

.53947 

.03463 

.57856 

.03789 

.61591 

.04130 

52  28 

.49917 

.03156 

.54014 

.03468 

.57920 

.03795 

.61652 

.04135 

50  29 

.49987 

.03161 

.54080 

.03474 

.57984 

.03800 

.61713 

.04141 

s    ' 

lh  22m    20° 

lh  jgflM        21° 

lh  30™    22° 

lh  34m    23° 

0  30 

8.50056 

0.03166 

8.54147 

0.03479 

8.58047 

0.03806 

8.61773 

0.04147 

4  31 

.50126 

.03171 

.54214 

.03484 

.58111 

.03812 

.61834 

.04153 

8  32 

.50196 

.03177 

.54280 

.03490 

.58174 

.03817 

.61895 

.04159 

.72  33 

.50266 

.03182 

.54346 

.03495 

.58238 

.03823 

.61955 

.04164 

.70  34 

.50335 

.03187 

.54413 

.03500 

.58301 

.03828 

.62016 

.04170 

20  35 

8.50405 

0.03192 

8.54479 

0.03506 

8.58364 

0.03834 

8.62077 

0.04176 

24  36 

.50475 

.03197 

.54545 

.03511 

.58427 

.03839 

.62137 

.04182 

28  37 

.50544 

.03202 

.54612 

.03517 

.58491 

.03845 

.62197 

.04188 

32  38 

.50614 

.03207 

.54678 

.03522 

.58554 

.03851 

.62258 

.04194 

30  39 

.50683 

.03212 

.54744 

.03527 

.58617 

.03856 

.62318 

.04199 

40  40 

8.50752 

0.03218 

8.54810 

0.03533 

8.58680 

0.03862 

8.62379 

0.04205 

44  41 

.50821 

.03223 

.54876 

.03538 

.58743 

.03867 

.62439 

.04211 

48  42 

.50891 

.03228 

.54942 

.03543 

.58806 

.03873 

.62499 

.04217 

52  43 

.50960 

.03233 

.55008 

.03549 

.58869 

.03879 

.62559 

.04223 

50  44 

.51029 

.03238 

.55073 

.03554 

.58932 

.03884 

.62619 

.04229 

s    ' 

lh  23m    20° 

lh  27m    21° 

lh  sim    22° 

lh  35m    23° 

0  45 

8.51098 

0.03243 

8.55139 

0.03560 

8.58994 

0.03890 

8.62680 

0.04234 

4  46 

.51167 

.03248 

.55205 

.03565 

.59057 

.03896 

.62740 

.04240 

5  47 

.51236 

.03254 

.55271 

.03570 

.59120 

.03901 

.62800 

.04246 

J2  48 

.51305 

.03259 

.55336 

.03576 

.59183 

.03907 

.62860 

.04252 

16  49 

.51374 

.03264 

.55402 

.03581 

.59245 

.03912 

.62919 

.04258 

20  50 

8.51442 

0.03269 

8.55467 

0.03587 

8.59308 

0.03918 

8.62979 

0.04264 

24  51 

.51511 

.03274 

.55533 

.03592 

.59370 

.03924 

.63039 

.04270 

28  52 

.51580 

.03279 

.55598 

.03597 

.59433 

.03929 

.63099 

.04276 

32  53 

.51648 

.03285 

.55664 

.03603 

.59495 

.03935 

.63159 

.04281 

30  54 

.51717 

.03290 

.55729 

.03608 

.59558 

.03941 

.63218 

.04287 

40  55 

8.51785 

0.03295 

8.55794 

0.03614 

8.59620 

0.03946 

8.63278 

0.04293 

44  56 

.51854 

.03300 

.55859 

.03619 

.59682 

.03952 

.63338 

.04299 

48  57 

.51922 

.03305 

.55925 

.03624 

.59745 

.03958 

.63397 

.04305 

52  58 

.51990 

.03311 

.55990 

.03630 

.59807 

.03963 

.63457 

.04311 

50  59 

.52058 

.03316 

.56055 

.03635 

.59869 

.03969 

.63516 

.04317 

00  60 

8.52127 

0.03321 

8.56120 

0.03641 

8.59931 

0.03975 

8.63576 

0.04323 

256 


Table  10.    Haversine  Table 


s   ' 

lh  gem    24° 

lh  40™    25° 

lh  44™    26° 

lh  48™    27° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.63576 

0.04323 

8.67067 

0.04685 

8.70418 

0.05060 

8.73637 

0.05450 

4   l 

.63635 

.04329 

.67124 

.04691 

.70472 

.05067 

.73690 

.05456 

8   2 

.63695 

.04335 

.67181 

.04697 

.70527 

.05073 

.73742 

.05463 

12   3 

.63754 

.04340 

.67238 

.04703 

.70582 

.05079 

.73795 

.05470 

16   4 

.63813 

.04346 

.67295 

.04709 

.70636 

.05086 

.73847 

.05476 

20   5 

8.63872 

0.04352 

8.67352 

0.04715 

8.70691" 

0.05092 

8.73900 

0.05483 

24   6 

.63932 

.04358 

.67409 

.04722 

.70745 

.05099 

,73952 

.05489 

28   7 

.63991 

.04364 

.67465 

.04728 

.70800 

.05105 

.74005 

.05496 

32   8 

.64050 

.04370 

.67522 

.04734 

.70854 

.05111 

.74057 

.05503 

36   9 

.64109 

.04376 

.67579 

.04740 

.70909 

.05118 

.74109 

.05509 

40  10 

8.64168 

0.04382 

8.67635 

0.04746 

8.70963 

0.05124 

8.74162 

0.05516 

44  11 

.64227 

.04388 

.67692 

.04752 

.71017 

.05131 

.74214 

.05523 

48  12 

.64286 

.04394 

.67748 

.04759 

.71072 

.05137 

.74266 

.05529 

52  13 

.64345 

.04400 

.67805 

.04765 

.71126 

.05144 

.74318 

.05536 

56  14 

.64404 

.04405 

.67861 

.04771 

.71180 

.05150 

.74371 

.05542 

s   ' 

lh  S7m    24° 

lh  41™    25° 

lh  45™    26° 

lh  49m    27° 

0  15 

8.64463 

0.04412 

8.67918 

0.04777 

8.71234 

0.05156 

8.74423 

0.05549 

4  16 

.64521 

.04418 

.67974 

.04783 

.71289 

.05163 

.74475 

.05556 

5  17 

.64580 

.04424 

.68030 

.04790 

.71343 

.05169 

.74527 

.05562 

12  18 

.64639 

.04430 

.68087 

.04796 

.71397 

.05176 

.74579 

.05569 

J0  19 

.64697 

.04436 

.68143 

.04802 

.71451 

.05182 

.74631 

.05576 

20  20 

8.64756 

0.04442 

8.68199 

0.04808 

8.71505 

0.05189 

8.74683 

0.05582 

24  21 

.64815 

.04448 

.68256 

.04815 

.71559 

.05195 

.74735 

.05589 

25  22 

.64873 

.04454 

.68312 

.04821 

.71613 

.05201 

.74787 

.05596 

G>ti)    OO 
t->^5    Ad 

.64932 

.04460 

.68368 

.04827 

.71667 

.05208 

.74839 

.05603 

55  24 

.64990 

.04466 

.68424 

.04833 

.71721 

.05214 

.74890 

.05609 

40  25 

8.65049 

0.04472 

8.68480 

0.04839 

8.71774 

0.05221 

8.74942 

0.05616 

44  26 

.65107 

.04478 

.68536 

.04846 

.71828 

.05227 

.74994 

.05623 

48  27 

.65165 

.04484 

.68592 

.04852 

.71882 

.05234 

.75046 

.05629 

52  28 

.65224 

.04490 

.68648 

.04858 

.71936 

.05240 

.75097 

.05636 

55  29 

.65282 

.04496 

.68704 

.04864 

.71989 

.05247 

.75149 

.05643 

s 

lh  ssm    24° 

lh  42™    25° 

lh  46m    26° 

lh  5Qm    27° 

0  30 

8.65340 

0.04502 

8.68760 

0.04871 

8.72043 

0.05253 

8.75201 

0.05649 

4  31 

.65398 

.04508 

.68815 

.04877 

.72097 

.05260 

.75252 

.05656 

8  32 

.65456 

.04514 

.68871 

.04883 

.72150 

.05266 

.75304 

.05663 

^2  33 

.65514 

.04520 

.68927 

.04890 

.72204 

.05273 

.75355 

.05670 

16  34 

.65572 

.04526 

.68983 

.04896 

.72257 

.05279 

.75407 

.05676 

20  35 

8.65630 

0.04532 

8.69038 

0.04902 

8.72311 

0.05286 

8.75458 

0.05683 

24  36 

.65688 

.04538 

.69094 

.04908 

.72364 

.05292 

.75510 

.05690 

28  37 

.65746 

.04544 

.69149 

.04915 

.72418 

.05299 

.75561 

.05697 

32  38 

.65804 

.04550 

.69205 

.04921 

.72471 

.05305 

.75613 

.05703 

36  39 

.65862 

.04556 

.69260 

.04927 

.72525 

.05312 

.75664 

.05710 

40  40 

8.65920 

0.04562 

8.69316 

0.04934 

8.72578 

0.05318 

8.75715 

0.05717 

44  41 

.65978 

.04569 

.69371 

.04940 

.72631 

.05325 

.75767 

.05724 

48  42 

.66035 

.04575 

.69427 

.04946 

.72684 

.05331 

.75818 

.05730 

52  43 

.66093 

.04581 

.69482 

.04952 

.72738 

.05338 

.75869 

.05737 

56  44 

.66151 

.04587 

.69537 

.04959 

.72791 

.05345 

.75920 

.05744 

s   ' 

lh  39m    24° 

lh  43™    25° 

lh  ^m      26° 

lh  51m    27° 

0  45 

8.66208 

0.04593 

8.69593 

0.04965 

8.72844 

0.05351 

8.75972 

O.C5751 

4  46 

.66266 

.04599 

.69648 

.04971 

.72897 

.05358 

.76023 

.05757 

5  47 

.66323 

.04605 

.69703 

.04978 

.72950 

.05364 

.76074 

.05764 

12  48 

.66381 

.04611 

.69758 

.04984 

.73003 

.05371 

.76125 

.05771 

iff  49 

.66438 

.04617 

.69814 

.04990 

.73056 

.05377 

.76176 

.05778 

20  50 

8.66496 

0.04623 

8.69869 

0.04997 

8.73109 

0.05384 

8.76227 

O.C5785 

24  51 

.66553 

.04629 

.69924 

.05003 

.73162 

.05390 

.76278 

.05791 

28  52 

.66610 

.04636 

.69979 

.05009 

.73215 

.05397 

.76329 

.05798 

32  53 

.66668 

.04642 

.70034 

.05016 

.73268 

.05404 

.76380 

.05805 

3S  54 

.66725 

.04648 

.70089 

.05022 

.73321 

.05410 

.76431 

.05812 

40  55 

8.66782 

0.04654 

8.70144 

0.05028 

8.73374 

0.05417 

8.76481 

0.05819 

44  56 

.66839 

.04660 

.70198 

.05035 

.73426 

.05423 

.76532 

.05825 

48  57 

.66896 

.04666 

.70253 

.05041 

.73479 

.05430 

.76583 

.05832 

52  58 

.66953 

.04672 

.70308 

.05048 

.73532 

.05436 

.76634 

.05839 

56  59 

.67010 

.04678 

.70363 

.05054 

.73584 

.05443 

.76684 

.05846 

(90  60 

8.67067 

0.04685 

8.70418 

005060 

8.73637 

0.05450 

8.76735 

0.05853 

Table  10.    Haversine  Table 


257 


s    ' 

1*  52™    28° 

lh  S6m      29° 

2h  Om    30° 

2h  jrn.     31° 

Hay. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.76735 

0.05853 

8.79720 

0.06269 

8.82599 

0.06699 

8.85380 

0.07142 

4   1 

.76786 

.05859 

.79769 

.06276 

.82646 

.06706 

.85425 

.07149 

8   2 

.76836 

.05866 

.79818 

.06283 

.82694 

.06713 

.85471 

.07157 

12   3 

.76887 

.05873 

.79866 

.06290 

.82741 

.06721 

.85516 

.07164 

16   4 

.76938 

.05880 

.79915 

.06297 

.82788 

.06728 

.85562 

.07172 

20   5 

8.76988 

0.05887 

8.79964 

0.06304 

8.82835 

0.06735 

8.85607 

0.07179 

24   6 

.77039 

.05894 

.80013 

.06311 

.82882 

.06742 

.85653 

.07187 

28   7 

.77089 

.05901 

.80061 

.06318 

.82929 

.06750 

.85698 

.07194 

32   8 

.77139 

.05907 

.80110 

.06326 

.82976 

.06757 

.85743 

.07202 

36   9 

.77190 

.05914 

.80158 

.06333 

.83023 

.06764 

.85789 

.07209 

40  10 

8.77240 

0.05921 

8.80207 

0.06340 

8.83069 

0.06772 

8.85834 

0.07217 

44  11 

.77291 

.05928 

.80256 

.06347 

.83116 

.06779 

.85879 

.07224 

48  12 

.77341 

.05935 

.80304 

.06354 

.83163 

.06786 

.85925 

.07232 

52  13 

.77391 

.05942 

.80353 

.06361 

.83210 

.06794 

.85970 

.07239 

56  14 

.77441 

.05949 

.80401 

.06368 

.83257 

.06801 

.86015 

.07247 

s   ' 

1*  53m    28° 

lh  57™    29° 

2h  I'*    30° 

9h  Qm      31° 

0  15 

8.77492 

0.05955 

8.80449 

0.06375 

8.83303 

0.06808 

8.86060 

0.07254 

4  16 

.77542 

.05962 

.80498 

.06382 

.83350 

.06816 

.86105 

.07262 

5  17 

.77592 

.05969 

.80546 

.06389 

.83397 

.06823 

.86151 

.07270 

12  18 

.77642 

.05976 

.80595 

.06397 

.83444 

.06830 

.86196 

.07277 

10  19 

.77692 

.05983 

.80643 

.06404 

.83490 

.06838 

.86241 

.07285 

20  20 

8,77742 

0.05990 

8.80691 

0.06411 

8.83537 

0.06845 

8.86286 

0.07292 

24  21 

.77792 

.05997 

.80739 

.06418 

.83583 

.06852 

.86331 

.07300 

2S  22 

.77842 

.06004 

.80788 

.06425 

.83630 

.06860 

.86376 

.07307 

32  23 

.77892 

.06011 

.80836 

.06432 

.83676 

.06867 

.86421 

.07315 

30  24 

.77942 

.06018 

.80884 

.06439 

.83723 

.06874 

.86466 

.07322 

40  25 

8.77992 

0.06024 

8.80932 

0.06446 

8.83769 

0.06882 

8.86511 

0.07330 

44  26 

.78042 

.06031 

.80980 

.06454 

.83816 

.06889 

.86556 

.07338 

45  27 

.78092 

.06038 

.81028 

.06461 

.83862 

.06896 

.86600 

.07345 

»2  28 

.78142 

.06045 

.81076 

.06468 

.83909 

.06904 

.86645 

.07353 

50  29 

.78191 

.06052 

.81124 

.06475 

.83955 

.06911 

.86690 

.07360 

s    ' 

1*  54m    28° 

lh  ggm      29° 

2h  2"1    30° 

gh  Qm     31° 

0  30 

8.78241 

0.06059 

8.81172 

0.06482 

8.84002 

0.06919 

8.86735 

0.07368 

4  31 

.78291 

.06066 

.81220 

.06489 

.84048 

.06926 

.86780 

.07376 

8  32 

.78341 

.06073 

.81268 

.06497 

.84094 

.06933 

.86825 

.07383 

12  33 

.78390 

.06080 

.81316 

.06504 

.84140 

.06941 

.86869 

.07391 

10  34 

.78440 

.06087 

.81364 

.06511 

.84187 

.06948 

.86914 

.07398 

20  35 

8.78490 

0.06094 

8.81412 

0.06518 

8.84233 

0.06956 

8.86959 

0.07406 

24  36 

.78539 

.06101 

.81460 

.06525 

.84279 

.06963 

.87003 

.07414 

28  37 

.78589 

.06108 

.81508 

.06532 

.84325 

.06970 

.87048 

.07421 

32  38 

.78638 

.06115 

.81555 

.06540 

.84371 

.06978 

.87093 

.07429 

S£  39 

.78688 

.06122 

.81603 

.06547 

.84417 

.06985 

.87137 

.07437 

40  40 

8.78737 

0.06129 

8.81651 

0.06554 

8.84464 

0.06993 

8.87182 

0.07444 

44  41 

.78787 

.06136 

.81699 

.06561 

.84510 

.07000 

.87226 

.07452 

48  42 

.78836 

.06143 

.81746 

.06568 

.84556 

.07007 

.87271 

.07459 

52  43 

.78885 

.06150 

.81794 

.06576 

.84602 

.07015 

.87315 

.07467 

50  44 

.78935 

.06157 

.81841 

.06583 

.S404S 

07022 

.87360 

.07475 

s    ' 

lh  55m    28° 

lh  5gm    29° 

2h3m    30° 

2h  7m     31° 

0  45 

8.78984 

0.06164 

8.81889 

0.06590 

8.84694 

0.07030 

8.87404 

0.07482 

4  46 

.79033 

.06171 

.81937 

.06597 

.84740 

.07037 

.87448 

.07490 

5  47 

.79082 

.06178 

.81984 

.06605 

.84785 

.07045 

.87493 

.07498 

12  48 

.79132 

.06185 

.82032 

.06612 

.84831 

.07052 

.87537 

.07505 

10  49 

.79181 

.06192 

.82079 

.06619 

.84877 

.07059 

.87582 

.07513 

20  50 

8.79230 

0.06199 

8.82126 

0.06626 

8.84923 

0.07067 

8.87626 

0.07521 

24  51 

.79279 

.06206 

.82174 

.06633 

.84969 

.07074 

.87670 

.07528 

28  52 

.79328 

.06213 

.82221 

.06641 

.85015 

.07082 

.87714 

.07536 

32  53 

.79377 

.06220 

.82269 

.06648 

.85060 

.07089 

.87759 

.07544 

30  54 

.79426 

.06227 

.82316 

.06655 

.851t)6 

.07097 

.87803 

.07551 

40  55 

8.79475 

0.06234 

8.82363 

0.06662 

8.85152 

0.07104 

8.87847 

0.07559 

44  56 

.79524 

.06241 

.82410 

.06670 

.85197 

.07112 

.87891 

.07567 

48  57 

.79573 

.06248 

.82458 

.06677 

.85243 

.07119 

.87935 

.07574 

52  58 

.79622 

.06255 

.82505 

.066S4 

.85289 

.07127 

.87980 

.07582 

50  59 

.79671 

.06262 

.82552 

.06691 

.85334 

.07134 

.88024 

.07590 

00  60 

8.79720 

0.06269 

8.82599 

0.06699 

8.85380 

0.07142 

8.88068 

0.07598 

258 


Table  10.    Haversine  Table 


s    ' 

2h  8™    32° 

2h  12m    33° 

2h  16™    34° 

2^  20m    35° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.88068 

0.07598 

8.90668 

0.08066 

8.93187 

0.08548 

8.95628 

0.09042 

4   1 

.88112 

.07605 

.90711 

.08074 

.93228 

.08556 

.95668 

.09051 

8   2 

.88156 

.07613 

.90754 

.08082 

.93270 

.08564 

.95709 

.09059 

12   3 

.88200 

.07621 

.90796 

.08090 

.93311 

.08573 

.95749 

.09067 

16   4 

.88244 

.07628 

.90839 

.08098 

.93352 

.08581 

.95789 

.09076 

20   5 

8.88288 

0.07636 

8.90881 

0.08106 

8.93393 

0.08589 

8.95828 

0.09084 

24   6 

.88332 

.07644 

.90924 

.08114 

.93435 

.08597 

.95868 

.09093 

28   7 

.88375 

.07652 

.90966 

.08122 

.93476 

.08605 

.95908 

.09101 

32   8 

.88419 

.07659 

.91009 

.08130 

.93517 

.08613 

.95948 

.09109 

36   9 

.88463 

.07667 

.91051 

.08138 

.93558 

.08621 

.95988 

.09118 

40  10 

8.88507 

0.07675 

8.91094 

0.08146 

8.93599 

0.08630 

8.96028 

0.09126 

44  11 

.88551 

.07683 

.91136 

.08154 

.93640 

.08638 

.96068 

.09134 

48  12 

.88595 

.07690 

.9117& 

.08162 

.93681 

.08646 

.96108 

.09143 

52  13 

.88638 

.07698 

.91221 

.08170 

.93722 

.08654 

.96148 

.09151 

56  14 

.88682 

.07706 

.91263 

.08178 

.93764 

.08662 

.96187 

.09160 

s    ' 

2h  9m    32° 

2h  IS"1    33° 

2h  17m     34° 

2h  21m    35° 

0  15 

8.88726 

0.07714 

8.91306 

0.08186 

8.93805 

0.08671 

8.96227 

0.09168 

4  16 

.88769 

.07721 

.91348 

.08194 

.93846 

.08679 

.96267 

.09176 

5  17 

.88813 

.07729 

.91390 

.08202 

.93886 

.08687 

.96307 

.09185 

12  18 

.88857 

.07737 

.91432 

.08210 

.93927 

.08695 

.96346 

.09193 

iff  19 

.88900 

.07745 

.91475 

.08218 

.93968 

.08703 

.96386 

.09202 

20  20 

8.88944 

0.07752 

8.91517 

0.08226 

8.94009 

0.08711 

8.96426 

0.09210 

24  21 

.88988 

.07760 

.91559 

.08234 

.94050 

.08720 

.96465 

.09218 

25  22 

.89031 

.07768 

.91601 

.08242 

.94091 

.08728 

.96505 

.09227 

32  23 

.89075 

.07776 

.91643 

.08250 

.94132 

.08736 

.96545 

.09235 

30  24 

.89118 

.07784 

.91685 

.08258 

.94173 

.08744 

.96584 

.09244 

40  25 

8.89162 

0.07791 

8.91728 

0.08266 

8.94213 

0.08753 

8.96624 

0.09252 

44  26 

.89205 

.07799 

.91770 

.08274 

.94254 

.08761 

.96663 

.09260 

48  27 

.89248 

.07807 

.91812 

.08282 

.94295 

.08769 

.96703 

.09269 

52  28 

.89292 

.07815 

.91854 

.08290 

.94336 

.08777 

.96742 

.09277 

50  29 

.89335 

.07823 

.91896 

.08298 

.94376 

.08785 

.96782 

.09286 

s    ' 

%h  iQm    32° 

2h  14m    33° 

&  18m    34° 

2*  22m    35° 

0  30 

8.89379 

0.07830 

8.91938 

0.08306 

8.94417 

0.08794 

8.96821 

0.09294 

4  31 

.89422 

.07838 

.91980 

.08314 

.94458 

.08802 

.96861 

.09303 

8  32 

.89465 

.07846 

.92022 

.08322 

.94498 

.08810 

.96900 

.09311 

12  33 

.89509 

.07854 

.92064 

.08330 

.94539 

.08818 

.96940 

.09320 

10  34 

.89552 

.07862 

.92105 

.08338 

.94580 

.08827 

.96979 

.09328 

20  35 

8.89595 

0.07870 

8.92147 

0.08346 

8.94620 

0.08835 

8.97018 

0.09337 

24  36 

.89638 

.07877 

.92189 

.08354 

.94661 

.08843 

.97058 

.09345 

28  37 

.89681 

.07885 

.92231 

.08362 

.94701 

.08851 

.97097 

.09353 

32  38 

.89725 

.07893 

.92273 

.08370 

.94742 

.08860 

.97136 

.09362 

30  39 

.89768 

.07901 

.92315 

.08378 

.94782 

.08868 

.97176 

.09370 

40  40 

8.89811 

0.07909 

8.92356 

0.08386 

8.94823 

0.08876 

8.97215 

0.09379 

44  41 

.89854 

.07917 

.92398 

.08394 

.94863 

.08885 

.97254 

.09387 

48  42 

.89897 

.07924 

.92440 

.08402 

.94904 

.08893 

.97294 

.09396 

52  43 

.89940 

.07932 

.92482 

.08410 

.94944 

.08901 

.97333 

.09404 

50  44 

.89983 

.07940 

.92523 

.08418 

.94985 

.08909 

.97372 

.09413 

s    ' 

2^  llm    32° 

%h  IQm      33° 

2h  19m    34° 

2h  23™    35° 

0  45 

8.90026 

0.07948 

8.92565 

0.08427 

8.95025 

0.08918 

8.97411 

0.09421 

4  46 

.90069 

.07956 

.92607 

.08435 

.95065 

.08926 

.97450 

.09430 

5  47 

.90112 

.07964 

.92648 

.08443 

.95106 

.08934 

.97489 

.09438 

12  48 

.90155 

.07972 

.92690 

.08451 

.95146 

.08943 

.97529 

.09447 

16  49 

.90198 

.07980 

.92731 

.08459 

.95186 

.08951 

.97568 

.09455 

20  50 

8.90241 

0.07987 

8.92773 

0.08467 

8.95227 

0.08959 

8.97607 

0.09464 

24  51 

.90284 

.07995 

.92814 

.08475 

.95267 

.08967 

.97646 

.09472 

28  52 

.90326 

.08003 

.92856 

.08483 

.95307 

.08976 

.97685 

.09481 

32  53 

.90369 

.08011 

.92897 

.08491 

.95347 

.08984 

.97724 

.09489 

30  54 

.90412 

.08019 

.92939 

.08499 

.95388 

.08992 

.97763 

.09498 

40  55 

8.90455 

0.08027 

8.92980 

0.08508 

8.95428 

0.09001 

8.97802 

0.09506 

44  56 

.90498 

.08035 

.93022 

.08516 

.95468 

.09009 

.97841 

.09515 

48  57 

.90540 

.08043 

.93063 

.08524 

.95508 

.09017 

.97880 

.09524 

52  58 

.90583 

.08051 

.93104 

.08532 

.95548 

.09026 

.97919 

.09532 

50  59 

.90626 

.08059 

.93146 

.08540 

.95588 

.09034 

.97958 

.09541 

00  60 

8.90668 

0.08066 

8.93187 

0.08548 

8.95628 

0.09042 

8.97997 

0.09549 

Table  10.    Haversine  Table 


259 


s    ' 

2*  24m    36° 

2*  28™    37° 

2*  32™    38° 

2*  36m    39° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

8.97997 

0.09549 

9.00295 

0.10068 

9.02528 

0.10599 

9.04699 

0.11143 

4   1 

.98035 

.09558 

.00333 

.10077 

.02565 

.10608 

.04735 

.11152 

8   2 

.98074 

.09566 

.00371 

.10086 

.02602 

.10617 

.04770 

.11161 

12   3 

.98113 

.09575 

.00408 

.10095 

.02638 

.10626 

.04806 

.11170 

16   4 

.98152 

.09583 

.00446 

.10103 

.02675 

.10635 

.04842 

.11179 

20   5 

8.98191 

0.09592 

9.00484 

0.10112 

9.02712 

0.10644 

9.04877 

0.11189 

24   6 

.98229 

.09601 

.00522 

.10121 

.02748 

.10653 

.04913 

.11198 

28   7 

.98268 

.09609 

.00559 

.10130 

.02785 

.10662 

.04948 

.11207 

32   8 

.98307 

.09618 

.00597 

.10138 

.02821 

.10671 

.04984 

.11216 

36   9 

.98346 

.09626 

.00634 

.10147 

.02858 

.10680 

.05019 

.11225 

40  10 

8.98384 

0.09635 

9.00672 

0.10156 

9.02894 

0.10689 

9.05055 

0.11234 

44  11 

.98423 

.09643 

.00710 

.10165 

.02931 

.10698 

.05090 

.11244 

48  12 

.98462 

.09652 

.00747 

.10174 

.02967 

.10707 

.05126 

.11253 

52  13 

.98500 

.09661 

.00785 

.10182 

.03004 

.10716 

.05161 

.11262 

56  14 

.98539 

.09669 

.00822 

.10191 

.03040 

.10725 

.05197 

.11271 

s    ' 

36° 

2h  2gm     37° 

2h  33m      38° 

2h  37m    39° 

0  15 

8.98578 

0.09678 

9.00860 

0.10200 

9,03077 

0.10734 

9.05232 

0.11280 

4  16 

.98616 

.09686 

.00897 

.10209 

.03113 

.10743 

.05268 

.11290 

5  17 

.98655 

.09695 

.00935 

.10218 

.03150 

.10752 

.05303 

.11299 

12  18 

.98693 

.09704 

.00972 

.10226 

.03186 

.10761 

.05339 

.11308 

.70  19 

.98732 

.09712 

.01009 

.10235 

.03222 

.10770 

.05374 

.11317 

20  20 

8.98770 

0.09721 

9.01047 

0.10244 

9.03259 

0.10779 

9.05409 

0.11326 

24  21 

.98809 

.09729 

.01084 

.10253 

.03295 

.10788 

.05445 

.11336 

25  22 

.98847 

.09738 

.01122 

.10262 

.03331 

.10797 

.05480 

.11345 

32  23 

.98886 

.09747 

.01159 

.10270 

.03368 

.10806 

.05515 

.11354 

36  24 

.98924 

.09755 

.01196 

.10279 

.03404 

.10815 

.05551 

.11363 

40  25 

8.98963 

0.09764 

9.01234 

0.10288 

9.03440 

0.10824 

9.05586 

0.11373 

44  26 

.99001 

.09773 

.01271 

.10297 

.03476 

.10833 

.05621 

.11382 

45  27 

.99039 

.09781 

.01308 

.10306 

.03513 

.10842 

.05656 

.11391 

52  28 

.99078 

.09790 

.01345 

.10315 

.03549 

.10851 

.05692 

.11400 

56  29 

.99116 

.09799 

.01383 

.10323 

.03585 

.10861 

.05727 

.11410 

s    ' 

36° 

2h  SQ™    37° 

2*  34m    38° 

2*  38™    39° 

0  30 

8.99154 

0.09807 

9.01420 

0.10332 

9.03621 

0.10870 

9.05762 

0.11419 

4  31 

.99193 

.09816 

.01457 

.10341 

.03657 

.10879 

.05797 

.11428 

8  32 

.99231 

.09824 

.01494 

.10350 

.03694 

.10888 

.05832 

.11437 

12  33 

.99269 

.09833 

.01531 

.10359 

.03730 

.10897 

.05867 

.11447 

10  34 

.99307 

.09842 

.01569 

.10368 

.03766 

.10906 

.05903 

.11456 

20  35 

8.99346 

0.09850 

9.01606 

0.10377 

9.03802 

0.10915 

9.05938 

0.11465 

24  36 

.99384 

.09859 

.01643 

.10386 

.03838 

.10924 

.05973 

.11474 

28  37 

.99422 

.09868 

.01680 

.10394 

.03874 

.10933 

.06008 

.11484 

32  38 

.99460 

.09876 

.01717 

.10403 

.03910 

.10942 

.06043 

.11493 

30  39 

.99498 

.09885 

.01754 

.10412 

.03946 

.10951 

.06078 

.11502 

40  40 

8.99536 

0.09894 

9.01791 

0.10421 

9.03982 

0.10960 

9.06113 

0.11511 

44  41 

.99575 

.09903 

.01828 

.10430 

.04018 

.10969 

.06148 

.11521 

48  42 

.99613 

.09911 

.01865 

.10439 

.04054 

.10978 

.06183 

.11530 

52  43 

.99651 

.09920 

.01902 

.10448 

.04090 

.10988 

.06218 

.11539 

50  44 

.99689 

.09929 

.01939 

.10457 

.04126 

.10997 

.06253 

.11549 

s 

%h  27m    36° 

2h  sim    37° 

2h  35m    38° 

2h  39™    39° 

0  45 

8.99727 

0.09937 

9.01976 

0.10466 

9.04162 

0.11006 

9.06288 

0.11558 

4  46 

.99765 

.09946 

.02013 

.10474 

.04198 

.11015 

.06323 

.11567 

8  47 

.99803 

.09955 

.02050 

.10483 

.04234 

.11024 

.06358 

.11577 

12  48 

.99841 

.09963 

.02087 

.10492 

.04270 

.11033 

.06393 

.11586 

16  49 

.99879 

.09972 

.02124 

.10501 

.04306 

.11042 

.06428 

.11595 

20  50 

8.99917 

0.09981 

9.02161 

0.10510 

9.04341 

0.11051 

9.06462 

0.11604 

24  51 

.99955 

.09990 

.02197 

.10519 

.04377 

.11060 

.06497 

.11614 

28  52 

.99993 

.09998 

.02234 

.10528 

.04413 

.11070 

.06532 

.11623 

32  53 

9.00031 

.10007 

.02271 

.10537 

.04449 

.11079 

.06567 

.11632 

30  54 

.00068 

.10016 

.02308 

.10546 

.04485 

.11088 

.06602 

.11642 

40  55 

9.00106 

0.10025 

9.02345 

0.10555 

9.04520 

0.11097 

9.06637 

0.11651 

44  56 

.00144 

.10033 

.02381 

.10564 

.04556 

.11106 

.06671 

.11660 

48  57 

.00182 

.10042 

.02418 

.10573 

.04592 

.11115 

.06706 

.11670 

52  58 

.00220 

.10051 

.02455 

.10582 

.04628 

.11124 

.06741 

.11679 

50  59 

.00258 

.10059 

.02492 

.10591 

.04663 

.11134 

.06776 

.11688 

(10  60 

9.00295 

0.10068 

9.02528 

0.10599 

9.04699 

0.11143 

9.06810 

0.11698 

260 


Table  10.    Haversine  Table 


s    ' 

2h  40m    40° 

2h  44m   41° 

2h  48m    42° 

&  52™    43° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.06810 

0.11698 

9.08865 

0.12265 

9.10866 

0.12843 

9.12815 

0.13432 

4   1 

.06845 

.11707 

.08899 

.12274 

.10899 

.12852 

.12847 

.13442 

8   2 

.06880 

.11716 

.08933 

.12284 

.10932 

.12862 

.12879 

.13452 

12   3 

.06914 

.11726 

.08966 

.12293 

.10965 

.12872 

.12911 

.13462 

16   4 

.06949 

.11735 

.09000 

.12303 

.10997 

.12882 

.12943 

.13472 

20   5 

9.06984 

0.11745 

9.09034 

0.12312 

9.11030 

0.12891 

9.12975 

0.13482 

24   6 

.07018 

.11754 

.09068 

.12322 

.11063 

.12901 

.13007 

.13492 

28   7 

.07053 

.11763 

.09101 

.12331 

.11096 

.12911 

.13039 

.13502 

32   8 

.07088 

.11773 

.09135 

.12341 

.11129 

.12921 

.13071 

.13512 

36   9 

.07122 

.11782 

.09169 

.12351 

.11161 

.12930 

.13103 

.13522 

40  10 

3.07157 

0.11791 

9.09202 

0.12360 

9.11194 

0.12940 

9.13135 

0.13532 

44  11 

.07191 

.11801 

.09236 

.12370 

.11227 

.12950 

.13167 

.13542 

48  12 

.07226 

.11810 

.09269 

.12379 

.11260 

.12960 

.13199 

.13552 

52  13 

.07260 

.11820 

.09303 

.12389 

.11292 

.12970 

.13231 

.13562 

56  14 

.07295 

.11829 

.09337 

.12398 

.11325 

.12979 

.13263 

.13571 

s    ' 

2^  41™    40° 

%h  4.5™     41o 

2h  49m    42° 

2h  53™    43° 

0  15 

9.07329 

0.11838 

9.09370 

0.12408 

9.11358 

0.12989 

9.13295 

0.13581 

4  16 

.07364 

.11848 

.09404 

.12418 

.11391 

.12999 

.13326 

.13591 

5  17 

.07398 

.11857 

.09437 

.12427 

.11423 

.13009 

.13358 

.13601 

12  18 

.07433 

.11867 

.09471 

.12437 

.11456 

.13018 

.13390 

.13611 

^6  19 

.07467 

.11876 

.09504 

.12446 

.11489 

.13028 

.13422 

.13621 

20  20 

9.07501 

0.11885 

9.09538 

0.12456 

9.11521 

0.13038 

9.13454 

0.13631 

24  21 

.07536 

.11895 

.09571 

.12466 

.11554 

.13048 

.13486 

.13641 

2S  22 

.07570 

.11904 

.09605 

.12475 

.11586 

.13058 

.13517 

.13651 

32  23 

.07605 

.11914 

.09638 

.12485 

.11619 

.13067 

.13549 

.13661 

36  24 

.07639 

.11923 

.09672 

.12494 

.11652 

.13077 

.13581 

.13671 

40  25 

9.07673 

0.11933 

9.09705 

0.12504 

9.11684 

0.13087 

9.13613 

0.13681 

44  26 

.07708 

.11942 

.09739 

.12514 

.11717 

.13097 

.13644 

.13691 

4<S  27 

.07742 

.11951 

.09772 

.12523 

.11749 

.13107 

.13676 

.13701 

52  28 

.07776 

.11961 

.09805 

.12533 

.11782 

.13116 

.13708 

.13711 

56  29 

.07810 

.11970 

.09839 

.12543 

.11814 

.13126 

.13739 

.13721 

s    ' 

2h  42™    40° 

&  46™    41° 

2h  50™    42° 

2h  54m    43° 

0  30 

9.07845 

0.11980 

9.09872 

0.12552 

9.11847 

0.13136 

9.13771 

0.13731 

4  31 

.07879 

.11989 

.09905 

.12562 

.11879 

.13146 

.13803 

.13741 

8  32 

.07913 

.11999 

.09939 

.12572 

.11912 

.13156 

.13834 

.13751 

^  33 

.07947 

.12008 

.09972 

.12581 

.11944 

.13166 

.13866 

.13761 

J6  34 

.07981 

.12018 

.10005 

.12591 

.11977 

.13175 

.13898 

.13771 

20  35 

9.08016 

0.12027 

9.10039 

0.12600 

9.12009 

0.13185 

9.13929 

0.13781 

24  36 

.08050 

.12036 

.10072 

.12610 

.12041 

.13195 

.13961 

.13791 

28  37 

.08084 

.12046 

.10105 

.12620 

.12074 

.13205 

.13992 

.13801 

32  38 

.08118 

.12055 

.10138 

.12629 

.12106 

.13215 

.14024 

.13811 

36  39 

.08152 

.12065 

.10172 

.12639 

.12139 

.13225 

.14056 

.13822 

40  40 

9.08186 

0.12074 

9.10205 

0.12649 

9.12171 

0.13235 

9.14087 

0.13832 

44  41 

.08220 

.12084 

.10238 

.12658 

.12203 

.13244 

.14119 

.13842 

48  42 

.08254 

.12093 

.10271 

.12668 

.12236 

.13254 

.14150 

.13852 

52  43 

.08288 

.12103 

.10304 

.12678 

.12268 

.13264 

.14182 

.13862 

56  44 

.08323 

.12112 

.10337 

.12687 

.12300 

.13274 

.14213 

.13872 

s    ' 

2h  43™    40° 

2h  47>n     41° 

2h  51m    42° 

&  55™    43° 

0  45 

9.08357 

0.12122 

9.10371 

0.12697 

9.12332 

0.13284 

9.14245 

0.13882 

4  46 

.08391 

.12131 

.10404 

.12707 

.12365 

.13294 

.14276 

.13892 

5  47 

.08425 

.12141 

.10437 

.12717 

,12397 

.13304 

.14307 

.13902 

12  48 

.08459 

.12150 

.10470 

.12726 

.12429 

.13314 

.14339 

.13912 

16  49 

.08492 

.12160 

.10503 

.12736 

.12461 

.13323 

.14370 

.13922 

20  50 

9.08526 

0.12169 

9.10536 

0.12746 

9.12494 

0.13333 

9.14402 

0.13932 

24  51 

.08560 

.12179 

.10569 

.12755 

.12526 

.13343 

.14433 

.13942 

2«S  52 

.08594 

.12188 

.10602 

.12765 

.12558 

.13353 

.14465 

.13952 

32  -  53 

.08628 

.12198 

.10635 

.12775 

.12590 

.13363 

.14496 

.13962 

36  54 

.08662 

.12207 

.10668 

.12784 

.12622 

.13373 

.14527 

.13972 

40  55 

9.08696 

0.12217 

9.10701 

0.12794 

9.12655 

0.13383 

9.14559 

0.13983 

44  56 

.08730 

.12226 

.10734 

.12804 

.12687 

.13393 

.14590 

.13993 

48  57 

.08764 

.12236 

.10767 

.12814 

.12719 

.13403 

.14621 

.14003 

52  58 

.08797 

.12245 

.10800 

.12823 

.12751 

.13412 

.14653 

.14013 

-50  59 

.08831 

.12255 

.10833 

.12833 

.12783 

.13422 

.14684 

.14023 

60  60 

9.08865 

0.12265 

9.10866 

0.12843 

9.12815 

0.13432 

9.14715 

0.14033 

Table  10.    Haversine  Table 


261 


s    ' 

2*  56m 

44° 

3hOm 

45° 

3h4m 

46° 

3h  8m 

47° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.14715 

0.14033 

9.16568 

0.14645 

9.18376 

0.15267 

9.20140 

0.15900 

4 

.14746 

.14043 

.16598 

.14655 

.18405 

.15278 

.20169 

.15911 

8 

.14778 

.14053 

.16629 

.14665 

.18435 

.15288 

.20198 

.15921 

12 

.14809 

.14063 

.16659 

.14676 

.18465 

.15298 

.20227 

.15932 

16 

.14840 

.14073 

.16690 

.14686 

.18495 

.15309 

.20256 

.15943 

20 

9.14871 

0.14084 

9.16720 

0.14696 

9.18524 

0.15319 

9.20285 

0.15953 

24 

.14902 

.14094 

.16751 

.14706 

.18554 

.15330 

.20314 

.15964 

28   7 

.14934 

.14104 

.16781 

.14717 

.18584 

.15340 

.20343 

.15975 

32   8 

.14965 

.14114 

.16812 

.14727 

.18613 

.15351 

.20372 

.15985 

36   9 

.14996 

.14124 

.16842 

.14737 

.18643 

.15361 

.20401 

.15996 

40  10 

9.15027 

0.14134 

9.16872 

0.14748 

9.18673 

0.15372 

9.20430 

0.16007 

44  11 

.15058 

.14144 

.16903 

.14758 

.18702 

.15382 

.20459 

.16017 

48  12 

.15089 

.14154 

.16933 

.14768 

.18732 

.15393 

.20488 

.16028 

52  13 

.15120 

.14165 

.16963 

.14779 

.18762 

.15403 

.20517 

.16039 

56  14 

.15152 

.14175 

.16994 

.14789 

.18791 

.15414 

.20546 

.16049 

s    ' 

2h  57m 

44° 

3h  jm 

45° 

&g* 

46° 

3h  9m 

47° 

0  15 

9.15183 

0.14185 

9.17024 

0.14799 

9.18821 

0.15424 

9.20574 

0.16060 

4  '16 

.15214 

.14195 

.17054 

.14810 

.18850 

.15435 

.20603 

.16071 

8  17 

.15245 

.14205 

.17085 

.14820 

.18880 

.15445 

.20632 

.16081 

J£  18 

.15276 

.14215 

.17115 

.14830 

.18909 

.15456 

.20661 

.16092 

16  19 

.15307 

.14226 

.17145 

.14841 

.18939 

.15466 

.20690 

.16103 

20  20 

9.15338 

0.14236 

9.17175 

0.14851 

9.18968 

0.15477 

9.20719 

0.16113 

24  21 

.15369 

.14246 

.17206 

.14861 

.18998 

.15487 

.20748 

.16124 

25  22 

.15400 

.14256 

.17236 

.14872 

.19027 

.15498 

.20776 

.16135 

32  23 

.15431 

.14266 

.17266 

.14882 

.19057 

.15509 

.20805 

.16146 

36  24 

.15462 

.14276 

.17296 

.14892 

.19086 

.15519 

.20834 

.16156 

40  25 

9.15493 

0.14287 

9.17327 

0.14903 

9.19116 

0.15530 

9.20863 

0.16167 

44  26 

.15524 

.14297 

.17357 

.14913 

.19145 

.15540 

.20891 

.16178 

4<S  27 

.15555 

.14307 

.17387 

.14923 

.19175 

.15551 

.20920 

.16188 

52  28 

.15585 

.14317 

.17417 

.14934 

.19204 

.15561 

.20949 

.16199 

56  29 

.15616 

.14327 

.17447 

.14944 

.19234 

.15572 

.20978 

.16210 

s    ' 

2h  58m 

44° 

3h  2m 

45° 

3h  6™ 

46° 

3h  iom 

47° 

0  30 

9.15647 

0.14337 

9.17477 

0.14955 

9.19263 

0.15582 

9.21006 

0.16220 

4  31 

.15678 

.14348 

.17507 

.14965 

.19292 

.15593 

.21035 

.16231 

5  32 

.15709 

.14358 

.17538 

.14975 

.19322 

.15603 

.21064 

.16242 

12  33 

.15740 

.14368 

.17568 

.14986 

.19351 

.15614 

.21092 

.16253 

^£  34 

.15771 

.14378 

.17598 

.14996 

.19381 

.15625 

.21121 

.16263 

20  35 

9.15802 

0.14388 

9.17628 

0.15006 

9.19410 

0.15635 

9.21150 

0.16274 

24  36 

.15832 

.14399 

.17658 

.15017 

.19439 

.15646 

.21178 

.16285 

25  37 

.15863 

.14409 

.17688 

.15027 

.19469 

.15656 

.21207 

.16296 

32  38 

.15894 

.14419 

.17718 

.15038 

.19498 

.15667 

.21236 

.16306 

SS  39 

.15925 

.14429 

.17748 

.15048 

.19527 

.15677 

.21264 

.16317 

40  40 

9.15955 

0.14440 

9.17778 

0.15058 

9.19557 

0.15688 

9.21293 

0.16328 

44  41 

.15986 

.14450 

.17808 

.15069 

.19586 

.15699 

.21322 

.16339 

45  42 

.16017 

.14460 

.17838 

.15079 

.19615 

.15709 

.21350 

.16349 

52  43 

.16048 

.14470 

.17868 

.15090 

.19644 

.15720 

.21379 

.16360 

flS  44 

.16078 

.14480 

.17898 

.15100 

.19674 

.15730 

.21407 

.16371 

s    ' 

2h  59m 

44° 

3h  gm 

45° 

3h  7m 

46° 

3*  llm 

47° 

0  45 

9.16109 

0.14491 

9.17928 

0.15110 

9.19703 

0.15741 

9.21436 

0.16382 

4  46 

.16140 

.14501 

.17958 

.15121 

.19732 

.15751 

.21464 

.16392 

8  47 

.16170 

.14511 

.17988 

.15131 

.19761 

.15762 

.21493 

.16403 

J£  48 

.16201 

.14521 

.18018 

.15142 

.19790 

.15773 

.21521 

.16414 

16  49 

.16232 

.14532 

.18048 

.15152 

.19820 

.15783 

.21550 

.16425 

20  50 

9.16262 

0.14542 

9.18077 

0.15163 

9.19849 

0.15794 

9.21578 

0.16436 

24  51 

.16293 

.14552 

.18107 

.15173 

.19878 

.15804 

.21607 

.16446 

28  52 

.16324 

.14562 

.18137 

.15183 

.19907 

.15815 

.21635 

.16457 

32  53 

.16354 

.14573 

.18167 

.15194 

.19936 

.15826 

.21664 

.16468 

36  54 

.16385 

.14583 

.18197 

.15204 

.19965 

.15836 

.21692 

.16479 

40  55 

9.16415 

0.14593 

9.18227 

0.15215 

9.19995 

0.15847 

9.21721 

0.16489 

44  56 

.16446 

.14604 

.18256 

.15225 

.20024 

.15858 

.21749 

.16500 

48  57 

.16476 

.14614 

.18286 

.15236 

.20053 

.15868 

.21778 

.16511 

52  58 

.16507 

.14624 

.18316 

.15246 

.20082 

.15879 

.21806 

.16522 

56  59 

.16537 

.14634 

.18346 

.15257 

.20111 

.15889 

.21834 

.16533 

60  60 

9.16568 

0.14645 

9.18376 

0.15267 

9.20140 

0.15900 

9.21863 

0.16543 

262 


Table  10.    Haversine  Table 


s    ' 

3h  i2m    48° 

3h  i&n    49° 

3h  20m    50° 

3h  24m    51° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.21863 

0.16543 

9.23545 

0.17197 

9.25190 

0.17861 

9.26797 

0.18534 

4   1 

.21891 

.16554 

.23573 

.17208 

.25217 

.17872 

.26823 

.18545 

8   2 

.21919 

.16565 

.23601 

.17219 

.25244 

.17883 

.26850 

.18557 

12   3 

.21948 

.16576 

.23629 

.17230 

.25271 

.17894 

.26876 

.18568 

16   4 

.21976 

.16587 

.23656 

.17241 

.25298 

.17905 

.26903 

.18579 

20   5 

9.22004 

0.16598 

9.23684 

0.17252 

9.25325 

0.17916 

9.26929 

0.18591 

24   6 

.22033 

.16608 

.23712 

.17263 

.25352 

.17928 

.26956 

.18602 

28   7 

.22061 

.16619 

.23739 

.17274 

.25379 

.17939 

.26982 

.18613 

32   8 

.22089 

.16630 

.23767 

.17285 

.25406 

.17950 

.27008 

.18624 

36   9 

.22118 

.16641 

.23794 

.17296 

.25433 

.17961 

.27035 

.18636 

40  10 

9.22146 

0.16652 

9.23822 

0.17307 

9.25460 

0.17972 

9.27061 

0.18647 

44  11 

.22174 

.16663 

.23850 

.17318 

.25487 

.17983 

.27088 

.18658 

48  12 

.22202 

.16673 

.23877 

.17329 

.25514 

.17995 

.27114 

.18670 

52  13 

.22231 

.16684 

.23905 

.17340 

.25541 

.18006 

.27140 

.18681 

56  14 

.22259 

.16695 

.23932 

.17351 

.25568 

.18017 

.27167 

.18692 

s    ' 

3^  13m    48° 

3h  Ijm     49° 

3h  21m    50° 

3h  25m    51° 

0  15 

9.22287 

0.16706 

9.23960 

0.17362 

9.25595 

0.18028 

9.27193 

0.18704 

4  16 

.22315 

.16717 

.23988 

.17373 

.25622 

.18039 

.27219 

.18715 

S  17 

.22343 

.16728 

.24015 

.17384 

.25649 

.18050 

.27246   .18727 

12  18 

.22372 

.16738 

.24043 

.17395 

.25676 

.18062 

.27272   .18738 

^5  19 

.22400 

.16749 

.24070 

.17406 

.25703 

.18073 

.27298 

.18749 

20  20 

9.22428 

0.16760 

9.24098 

0.17417 

9.25729 

0.18084 

9.27325 

0.18761 

24  21 

.22456 

.16771 

.24125 

.17428 

.25756 

.18095 

.27351   .18772 

25  22 

.22484 

.16782 

.24153 

.17439 

.25783 

.18106 

.27377   .18783 

32  23 

.22512 

.16793 

.24180 

.17450 

.25810 

.18118 

.27403   .18795 

30  24 

.22540 

.16804 

.24208 

.17461 

.25837 

.18129 

.27430 

.18806 

40  25 

9.22569 

0.16815 

9.24235 

0.17472 

9.25864 

0.18140 

9.27456 

0.18817 

44  26 

.22597 

.16825 

.24263 

.17483 

.25891 

.18151 

.27482 

.18829 

45  27 

.22625 

.16836 

.24290 

.17494 

.25917 

.18162 

.27508 

.18840 

52  28 

.22653 

.16847 

.24317 

.17505 

.25944 

.18174 

.27535 

.18852 

55  29 

.22681 

.16858 

.24345 

.17517 

.25971 

.18185 

.27561 

.18863 

s    ' 

3h  Uf*     48° 

3h  ism    49° 

3h  22"1    50° 

3h  26™    51° 

0  30 

9.22709 

0.16869 

9.24372 

0.17528 

9.25998 

0.18196 

9.27587 

0.18874 

4  31 

.22737 

.16880 

.24400 

.17539 

.26025 

.18207 

.27613 

.18886 

8  32 

.22765 

.16891 

.24427 

.17550 

.26051 

.18219 

.27639 

.18897 

.72  33 

.22793 

.16902 

.24454 

.17561 

.26078 

.18230 

.27666 

.18908 

^5  34 

.22821 

.16913 

.24482 

.17572 

.26105 

.18241 

.27692 

.18920 

20  35 

9.22849 

0.16924 

9.24509 

0.17583 

9.26132 

0.18252 

9.27718 

0.18931 

24  36 

.22877 

.16934 

.24536 

.17594 

.26158 

.18263 

.27744 

.18943 

28  37 

.22905 

.16945 

.24564 

.17605 

.26185 

.18275 

.27770 

.18954 

32  38 

.22933 

.16956 

.24591 

.17616 

.26212 

.18286 

.27796 

.18965 

35  39 

.22961 

.16967 

.24618 

.17627 

.26238 

.18297 

.27822 

.18977 

40  40 

9.22989 

0.16978 

9.24646 

0.17638 

9.26265 

0.18308 

9.27848 

0.18988 

44  41 

.23017 

.16989 

.24673 

.17649 

.26292 

.18320 

.27875 

.19000 

48  42 

.23045 

.17000 

.24700 

.17661 

.26319 

.18331 

.27901 

.19011 

52  43 

.23073 

.17011 

.24728 

.17672 

.26345 

.18342 

.27927 

.19022 

55  44 

.23100 

.17022 

.24755 

.17683 

.26372 

.18353 

.27953 

.19034 

s    ' 

3h  I5m    48° 

gh  IQrn     490 

3h  ^S"1    50° 

gh  27m    51° 

0  45 

9.23128 

0.17033 

9.24782 

0.17694 

9.26398 

0.18365 

9.27979 

0.19045 

4  46 

.23156 

.17044 

.24809 

.17705 

.26425 

.18376 

.28005 

.19057 

1  47 

.23184 

.17055 

.24837 

.17716 

.26452 

.18387 

.28031 

.19068 

^2  48 

.23212 

.17066 

.24864 

.17727 

.26478 

.18399 

.28057 

.19080 

16  49 

.23240 

.17076 

.24891 

.17738 

.26505 

.18410 

.28083 

.19091 

20  50 

9.23268 

0.17087 

9.24918 

0.17749 

9.26532 

0.18421 

9.28109 

0.19102 

24  51 

.23295 

.17098 

.24945 

.17760 

.26558 

.18432 

.28135 

.19114 

25  52 

.23323 

.17109 

.24973 

.17772 

.26585 

.18444 

.28161 

.19125 

32  53 

.23351 

.17120 

.25000 

.17783 

.26611 

.18455 

.28187 

.19137 

36  54 

.23379 

.17131 

.25027 

.17794 

.26638 

.18466 

.28213 

.19148 

40  55 

9.23407 

0.17142 

9.25054 

0.17805 

9.26664 

0.18478 

9.28239 

0.19160 

44  56 

.23434 

.17153 

.25081 

.17816 

.26691 

.18489 

.28265 

.19171 

48  57 

.23462 

.17164 

.25108 

.17827 

.26717 

.18500 

.28291 

.19183 

52  53 

.23490 

.17175 

.25135 

.17838 

.26744 

.18511 

.28317 

.19194 

56  59 

.23518 

.17186 

.25163 

.17849 

.26770 

.18523 

.28342 

.19205 

50  60 

9.23545 

0.17197 

9.25190 

0.17861 

9.26797 

0.18534 

9.28368 

0.19217 

Table  10.    Haversine  Table 


263 


s    ' 

3*  28™    52° 

gh  sgm     530 

3*  36™    54° 

3^  40m    55° 

Hay. 

No. 

Hav. 

No. 

Hay. 

No. 

Hav. 

No. 

0   0 

9.28368 

0.19217 

9.29906 

0.19909 

9.31409 

0.20611 

9.32881 

0.21321 

4   1 

.28394 

.19228 

.29931 

.19921 

.31434 

.20623 

.32905 

.21333 

8   2 

.28420 

.19240 

.29956 

.19932 

.31459 

.20634 

.32930 

.21345 

12   3 

.28446 

.19251 

.29981 

.19944 

.31484 

.20646 

.32954 

.21357 

16   4 

.28472 

.19263 

.30007 

.19956 

.31508 

.20658 

.32978 

.21369 

20   5 

9.28498 

0.19274 

9.30032 

0.19967 

9.31533 

0.20670 

9.33002 

0.21381 

24   6 

.28524 

.19286 

.30057 

.19979 

.31558 

.20681 

.33027 

.21393 

28   7 

.28549 

.19297 

.30083 

.19991 

.31583 

.20693 

.33051 

.21405 

32   8 

.28575 

.19309 

.30108 

.20002 

.31607 

.20705 

.33075 

.21417 

36   9 

.28601 

.19320 

.30133 

.20014 

.31632 

.20717 

.33099 

.21429 

40  10 

9.28627 

0.19332 

9.30158 

0.20026 

9.31657 

0.20729 

9.33123 

0.21440 

44  11 

.28653 

.19343 

.30184 

.20037 

.31682 

.20740 

.33148 

.21452 

48  12 

.28679 

.19355 

.30209 

.20049 

.31706 

.20752 

.33172 

.21464 

52  13 

.28704 

.19366 

.30234 

.20060 

.31731 

.20764 

.33196 

.21476 

56  14 

.28730 

.19378 

.30259 

.20072 

.31756 

.20776 

.33220 

.21488 

s    ' 

3h  29m    52° 

3*  33™    53° 

3h  S7m    54° 

3h  41™    55° 

0  15 

9.28756 

0.19389 

9.30285 

0.20084 

9.31780 

0.20788 

9.33244 

0.21500 

4  16 

.28782 

.19401 

.30310 

.20095 

.31805 

.20799 

.33268 

.21512 

5  17 

.28807 

.19412 

.30335 

.20107 

.31830 

.20811 

.33292 

.21524 

12  18 

.28833 

.19424 

.30360 

.20119 

.31854 

.20823 

.33317 

.21536 

16  19 

.28859 

.19435 

.30385 

.20130 

.31879 

.20835 

.33341 

.21548 

20  20 

9.28885 

0.19447 

9.30410 

0.20142 

9.31903 

0.20847 

9.33365 

0.21560 

24  21 

.28910 

.19458 

.30436 

.20154 

.31928 

.20858 

.33389 

.21572 

25  22 

.28936 

.19470 

.30461 

.20165 

.31953 

.20870 

.33413 

.21584 

32  23 

.28962 

.19481 

.30486 

.20177 

.31977 

.20882 

.33437 

.21596 

36  24 

.28987 

.19493 

.30511 

.20189 

.32002 

.20894 

.33461 

.21608 

40  25 

9.29013 

0,19504 

9.30536 

0.20200 

9.32026 

0.20906 

9.33485 

0.21620 

44  26 

.29039 

.19516 

.30561 

.20212 

.32051 

.20918 

.33509 

.21632 

45  27 

.29064 

.19527 

.30586 

.20224 

.32076 

.20929 

.33533 

.21644 

52  28 

.29090 

.19539 

.30611 

.20235 

.32100 

.20941 

.33557 

.21656 

.56  29 

.29116 

.19550 

.30636 

.20247 

.32125 

.20953 

.33581 

.21668 

s    ' 

3h  SQ™    52° 

3*-  34m    53° 

3*  38m    54° 

3h  j^2m    55° 

0  30 

9.29141 

0.19562 

9.30662 

0.20259 

9.32149 

0.20965 

9.33605 

0.21680 

4  31 

.29167 

.19573 

.30687 

.20271 

.32174 

.20977 

.33629 

.21692 

8  32 

.29192 

.19585 

.30712 

.20282 

.32198 

.20989 

.33653 

.21704 

J2  33 

.29218 

.19597 

.30737 

.20294 

.32223 

.21000 

.33677 

.21716 

J6  34 

.29244 

.19608 

.30762 

.20306 

.32247 

.21012 

.33701 

.21728 

20  35 

9.29269 

0.19620 

9.30787 

0.20317 

9.32272 

0.21024 

9.33725 

0.21740 

24  36 

.29295 

.19631 

.30812 

.20329 

.32296 

.21036 

.33749 

.21752 

28  37 

.29320 

.19643 

.30837 

.20341 

.32321 

.21048 

.33773 

.21764 

32  38 

.29346 

.19654 

.30862 

.20352 

.32345 

.21060 

.33797 

.21776 

36  39 

.29371 

.19666 

.30887 

.20364 

.32370 

.21072 

.33821 

.21788 

40  40 

9.29397 

0.19677 

9.30912 

0.20376 

9.32394 

0.21083 

9.33845 

0.21800 

44  41 

.29422 

,19689 

.30937 

.20388 

.32418 

.21095 

.33869 

.21812 

48  42 

.29448 

.19701 

.30962 

.20399 

.32443 

.21107 

.33893 

.21824 

52  43 

.29473 

.19712 

.30987 

.20411 

.32467 

.21119 

.33917 

.21836 

56  44 

.29499 

.19724 

.31012 

.20423 

.32492 

.21131 

.33941 

.21848 

s    ' 

3h  31m    52° 

3*  35m    53° 

3h  39™    54° 

3*  48m    55° 

0  45 

9.29524 

0.19735 

9.31036 

0.20435 

9.32516 

0.21143 

9.33965 

0.21860 

4  46 

.29550 

.19747 

.31061 

.20446 

.32541 

.21155 

.33988 

.21872 

5  47 

.29575 

.19758 

.31086 

.20458 

.32565 

.21167 

.34012 

.21884 

J2  48 

.29601 

.19770 

.31111 

.20470 

.32589 

.21178 

.34036 

.21896 

16  49 

.29626 

.19782 

.31136 

.20481 

.32614 

.21190 

.34060 

.21908 

20  50 

9.29652 

0.19793 

9.31161 

0.20493 

9.32638 

0.21202 

9.34084 

0.21920 

24  51 

.29677 

.19805 

.31186 

.20505 

.32662 

.21214 

.34108 

.21932 

28  52 

.29703 

.19816 

.31211 

.20517 

.32687 

.21226 

.34132 

.21944 

32  53 

.29728 

.19828 

.31236 

.20528 

.32711 

.21238 

.34155 

.21956 

36  54 

.29753 

.19840 

.31260 

.20540 

.32735 

.21250 

.34179 

.21968 

40  55 

9.29779 

0.19851 

9.31285 

0.20552 

9.32760 

0.21262 

9.34203 

0.21980 

44  56 

.29804 

.19863 

.31310 

.20564 

.32784 

.21274 

.34227 

.21992 

48  57 

.29829 

.19874 

.31335 

.20575 

.32808 

.21235 

.34251 

.22004 

52  58 

.29855 

.19886 

.31360 

.20587 

.32833 

.21297 

.34274 

.22016 

56  59 

.29880 

.19898 

.31385 

.20599 

.32857 

.21309 

.34298 

.22028 

60  60 

9.29906 

0.19909 

9.31409 

0.20611 

9.32881 

0.21321 

9.34322 

0.22040 

264 


Table  10.    Haversine  Table 


1 

3h  44™    56° 

3^  48™    57° 

Sh  52™    58° 

3*  56™    59° 

HAT. 

No. 

Ilav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.34322 

0.22040 

9.35733 

0.22768 

9.37114 

0.23504 

9.38468 

0.24248 

4  1 

.34346 

.22052 

.35756 

.22780 

.37137 

.23516 

.38490 

.24261 

8   2 

.34369 

.22064 

.35779 

.22792 

.37160 

.23529 

.38512 

.24273 

12   3 

.34393 

.22077 

.35802 

.22805 

.37183 

.23541 

.38535 

.24286 

16   4 

.34417 

.22089 

.35826 

.22817 

.37205 

.23553 

.38557 

.24298 

20   5 

9.34441 

0.22101 

9.35849 

0.22829 

9.37228 

0.23566 

9.38579 

0.24310 

24   6 

.34464 

.22113 

.35872 

.22841 

.37251 

.23578 

.38602 

.24323 

.34488 

.22125 

.35895 

.22853 

.37274 

.23590 

.38624 

.24335 

82   8 

.34512 

.22137 

.35918 

.22866 

.37296 

.23603 

.38646 

.24348 

35   9 

.34535 

.22149 

.35942 

.22878 

.37319 

.23615 

.38668 

.24360 

40  10 

9.34559 

0.22161 

9.35965 

0.22890 

9.37342 

0.23627 

9.38691 

0.24373 

44  11 

.34583 

.22173 

.35988 

.22902 

.37364 

.23640 

.38713 

.24385 

45  12 

.34606 

.22185 

.36011 

.22915 

.37387 

.23652 

.38735 

.24398 

52  13 

.34630 

.22197 

.36034 

.22927 

.37410 

.23665 

.38757 

.24410 

55  14 

.34654 

.22209 

.36058 

.22939 

.37433 

.23677 

.38780 

.24423 

s 

3^  45m    56° 

3h  49™    57° 

Sh  53™    58° 

3h  57m     59° 

0  15 

9.34677 

0.22221 

9.36081 

0.22951 

9.37455 

0.23689 

9.38802 

0.24435 

4  16 

.34701 

.22234 

.36104 

.22964 

.37478 

.23702 

.38824 

.24448 

5  17 

.34725 

.22246 

.36127 

.22976 

.37501 

.23714 

.38846 

.24460 

12  18 

.34748 

.22258 

.36150 

.22988 

.37523 

.23726 

.38868 

.24473 

^5  19 

.34772 

.22270 

.36173 

.23000 

.37546 

.23739 

.38891 

.24485 

20  20 

9.34795 

0.22282 

9.36196 

0.23012 

9.37569 

0.23751 

9.38913 

0.24498 

24  21 

.34819 

.22294 

.36219 

.23025 

.37591 

.23764 

.38935 

.24510 

25  22 

.34843 

.22306 

.36243 

.23037 

.37614 

.23776 

.38957 

.24523 

32  23 

.34866 

.22318 

.36266 

.23049 

.37636 

.23788 

.38979 

.24535 

35  24 

.34890 

.22330 

.36289 

.23061 

.37659 

.23801 

.39002 

.24548 

40  25 

9.34913 

0.22343 

9.36312 

0.23074 

9.37682 

0.23813 

9.39024 

0.24560 

44  26 

.34937 

.22355 

.36335 

.23086 

.37704 

.23825 

.39046 

.24573 

45  27 

.34960 

.22367 

.36358 

.23098 

.37727 

.23838 

.39068 

.24586 

52  23 

.34984 

.22379 

.36381 

.23110 

.37749 

.23850 

.39090 

.24598 

55  29 

.35007 

.22391 

.36404 

.23123 

.37772 

.23863 

.39112 

.24611 

s   ' 

8*  46m    56° 

Sh  50m    57° 

&  54m    58° 

Sh  gs™    59° 

0  3D 

9.35031 

0.22403 

9.36427 

0.23135 

9.37794 

0.23875 

9.39134 

0.24623 

4  31 

.35054 

.22415 

.36450 

.23147 

.37817 

.23887 

.39156 

.24636 

8  32 

.35078 

.22427 

.36473 

.23160 

.37840 

.23900 

.39178 

.24648 

^2  33 

.35101 

.22440 

.36496 

.23172 

.37862 

.23912 

.39201 

.24661 

16  34 

.35125 

.22452 

.36519 

.23184 

.37885 

.23925 

.39223 

.24673 

20  35 

9.35148 

0.22464 

9.36542 

0.23196 

9.37907 

0.23937 

9.39245 

0.24686 

24  36 

.35172 

.22476 

.36565 

.23209 

.37930 

.23950 

.39267 

.24698 

28  37 

.35195 

.22488 

.36588 

.23221 

.37952 

.23962 

.39289 

.24711 

32  38 

.35219 

.22500 

.36611 

.23233 

.37975 

.23974 

.39311 

.24723 

36  39 

.35242 

.22512 

.36634 

.23246 

.37997 

.23987 

.39333 

.24736 

40  40 

9.35266 

0.22525 

9.36657 

0.23258 

9.38020 

0.23999 

9.39355 

0.24749 

44  41 

.35289 

.22537 

.36680 

.23270 

.38042 

.24012 

.39377 

.24761 

48  42 

.35312 

.22549 

.36703 

.23282 

.38065 

.24024 

.39399 

.24774 

52  43 

.35336 

.22561 

.36726 

.23295 

.38087 

.24036 

.39421 

.24786 

56  44 

.35359 

.22573 

.36749 

.23307 

.38110 

.24049 

.39443 

.24799 

s    ' 

3h  47m    56° 

3h  6im     570 

Sh  55™    58° 

Sh  QQm    59° 

0  45 

9.35383 

0.22585 

9.36772 

0.23319 

9.38132 

0.24061 

9.39465 

0.24811 

4  46 

.35406 

.22598 

.36794 

.23332 

.38154 

.24074 

.39487 

.24824 

5  47 

.35429 

.22610 

.36817 

.23344 

.38177 

.24086 

.39509 

.24836 

12  48 

.35453 

.22622 

.36840 

.23356 

.38199 

.24099 

.39531 

.24849 

^5  49 

.35476 

.22634 

.36863 

.23368 

.38222 

.24111 

.39553 

.24862 

20  50 

9.35500 

0.22646 

9.36886 

0.23381 

9.38244 

0.24124 

9.39575 

0.24874 

24  51 

.35523 

.22658 

.36909 

.23393 

.38267 

.24136 

.39597 

.24887 

28  52 

.35546 

.22671 

.36932 

.23405 

.38289 

.24148 

.39619 

.24899 

32  53 

.35570 

.22683 

.36955 

.23418 

.38311 

.24161 

.39641 

.24912 

35  54 

.35593 

.22695 

.36977 

.23430 

.38334 

.24173 

.39663 

.24924 

40  55 

9.35616 

0.22707 

9.37000 

0.23442 

9.38356 

0.24186 

9.39685 

0.24937 

44  56 

.35639 

.22719 

.37023 

.23455 

.38378 

.24198 

.39706 

.24950 

48  57 

.35663 

.22731 

.37046 

.23467 

.38401 

.24211 

.39728 

.24962 

52  58 

.35686 

.22744 

.37069 

.23479 

.38423 

.24223 

.39750 

.24975 

56  59 

.35709 

.22756 

.37091 

.23492 

.38445 

.24236 

.39772 

.24987 

50  60 

9.35733 

0.22768 

9.37114 

0.235G4 

9.38468 

0.24248 

9.39794  0.25000 

Table  10.    Haversine  Table 


265 


s    ' 

4*  #"»    60° 

4*  4m    61° 

4*  8m    62° 

4*  12m    63° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.39794 

0.25000 

9.41094 

0.25760 

9.42368 

0.26526 

9.43617 

0.27300 

4   1 

.39816 

.25013 

.41115 

.25772 

.42389 

.26539 

.43638 

.27313 

8   2 

.39838 

.25025 

.41137 

.25785 

.42410 

.26552 

.43658 

.27326 

12   3 

.39860 

.25038 

.41158 

.25798 

.42431 

.26565 

.43679 

.27339 

16   4 

.39881 

.25050 

.41180 

.25810 

.42452 

.26578 

.43699 

.27352 

20   5 

9.39903 

0.25063 

9.41201 

0.25823 

9.42473 

0.26591 

9.43720 

0.27365 

24   6 

.39925 

.25076 

.41222 

.25836 

.42494 

.26604 

.43741 

.27378 

28   7 

.39947 

.25088 

.41244 

.25849 

.42515 

.26616 

.43761 

.27391 

S#   8 

.39969 

.25101 

.41265 

.25861 

.42536 

.26629 

.43782 

.27404 

36   9 

.39991 

.25113 

.41287 

.25874 

.42557 

.26642 

.43802 

.27417 

40  10 

9.40012 

0.25126 

9.41308 

0.25887 

9.42578 

0.26655 

9.43823 

0.27430 

44  11 

.40034 

.25139 

.41329 

.25900 

.42599 

.26668 

.43843 

.27443 

48  12 

.40056 

.25151 

.41351 

.25912 

.42620 

.26681 

.43864 

.27456 

52  13 

.40078 

.25164 

.41372 

.25925 

.42641 

.26694 

.43884 

.27469 

56  14 

,40100 

.25177 

.41393 

.25938 

.42662 

.26706 

.43905 

.27482 

s   ' 

4*  lm    60° 

4h  5m    61° 

4*  9m    62° 

4*  13m    63° 

0  15 

9.40121 

0.25189 

9.41415 

0.25951 

9.42682 

0.26719 

9.43926 

0.27495 

4  16 

.40143 

.25202 

.41436 

.25963 

.42703 

.26732 

.43946 

.27508 

S  17 

.40165 

.25214 

.41457 

.25976 

.42724 

.26745 

.43967 

.27521 

12  18 

.40187 

25227 

.41479 

.25989 

.42745 

.26758 

.43987 

.27534 

/£  19 

.40208 

.25240 

.41500 

.26002 

.42766 

.26771 

.44008 

.27547 

20  20 

9.40230 

0.25252 

9.41521 

0.26014 

9.42787 

0.26784 

9.44028 

0.27560 

24  21 

.40252 

.25265 

.41543 

.26027 

.42808 

.26797 

.44048 

.27573 

25  22 

.40274 

.25278 

.41564 

.26040 

.42829 

.26809 

.44069 

.27586 

32  23 

.40295 

.25290 

.41585 

.26053 

.42850 

.26822 

.44089 

.27599 

3£  24 

.40317 

.25303 

.41606 

.26065 

.42870 

.26835 

.44110 

.27612 

40  25 

9.40339 

0.25316 

9.41628 

0.26078 

9.42891 

0.26848 

9.44130 

0.27625 

44  26 

.40360 

.25328 

.41649 

.26091 

.42912 

.26861 

.44151 

.27638 

48  27 

.40382 

.25341 

.41670 

.26104 

.42933 

.26874 

.44171 

.27651 

52  28 

.40404 

.25354 

.41692 

.26117 

.42954 

.26887 

.44192 

.27664 

5£  29 

.40425 

.25366 

.41713 

.26129 

.42975 

.26900 

.44212 

.27677 

s   ' 

4h  2"1    60° 

4*  6™    61° 

4*  10m   62° 

4*  I4m   63° 

0  30 

9.40447 

0.25379 

9.41734 

0.26142 

9.42996 

0.26913 

9.44232 

0.27690 

4  31 

.40469 

.25391 

.41755 

.26155 

.43016 

.26925 

.44253 

.27703 

8  32 

.40490 

.25404 

.41776 

.26168 

.43037 

.26938 

.44273 

.27716 

J2  33 

.40512 

.25417 

.41798 

.26180 

.43058 

.26951 

.44294 

.27729 

J0  34 

.40534 

.25429 

.41819 

.26193 

.43079 

.26964 

.44314 

.27742 

20  35 

9.40555 

0.25442 

9.41840 

0.26206 

9.43100 

0.26977 

9.44334 

0.27755 

24  36 

.40577 

.25455 

.41861 

.26219 

.43120 

.26990 

.44355 

.27768 

28  37 

.40599 

.25467 

.41882 

.26232 

.43141 

.27003 

.44375 

.27781 

32  38 

.40620 

.25480 

.41904 

.26244 

.43162 

.27016 

.44396 

.27794 

3£  39 

.40642 

.25493 

.41925 

.26257 

.43183 

.27029 

.44416 

.27807 

40  40 

9.40663 

0.25506 

9.41946 

0.26270 

9.43203 

0.27042 

9.44436 

0.27820 

44  41 

.40685 

.25518 

.41967 

.26283 

.43224 

.27055 

.44457 

.27833 

48  42 

.40707 

.25531 

.41988 

.26296 

.43245 

.27068 

.44477 

.27846 

52  43 

.40728 

.25544 

.42009 

.26308 

.43266 

.27080 

.44497 

.27859 

50  44 

.40750 

.25556 

.42031 

.26321 

.43286 

.27093 

.44518 

.27873 

s   ' 

4*3™    60° 

4h  7m    61° 

4h  llm    62° 

4*  15m    63° 

0  45 

9.40771 

0.25569 

9.42052 

0.26334 

9.43307 

0.27106 

9.44538 

0.27886 

4  46 

.40793 

.25582 

.42073 

.26347 

.43328 

.27119 

.44558 

.27899 

S  47 

.40814 

.25594 

.42094 

.26360 

.43348 

.27132 

.44579 

.27912 

^2  48 

.40836 

.25607 

.42115 

.26372 

.43369 

.27145 

.44599 

.27925 

16  49 

.40858 

.25620 

.42136 

.26385 

.43390 

.27158 

.44619 

.27938 

20  50 

9.40879 

0.25632 

9.42157 

0.26398 

9.43411 

0.27171 

9.44639 

0.27951 

24  51 

.40900 

.25645 

.42178 

.26411 

.43431 

.27184 

.44660 

.27964 

28  52 

.40922 

.25658 

.42199 

.26424 

.43452 

.27197 

.44680 

.27977 

32  53 

.40943 

.25671 

.42221 

.26437 

.43473 

.27210 

.44700 

.27990 

36  54 

.40965 

.25683 

.42242 

.26449 

.43493 

.27223 

.44721 

.28003 

40  55 

9.40986 

0.25696 

9.42263 

0.26462 

9.43514 

0.27236 

9.44741 

0.28016 

44  66 

.41008 

.25709 

.42284 

.26475 

.43535 

.27249 

.44761 

.28029 

45  57 

.41029 

.25721 

.42305 

.26488 

.43555 

.27262 

.44781 

.28042 

52  58 

.41051 

.25734 

.42326 

.26501 

.43576 

.27275 

.44801 

.28055 

56  59 

.41072 

.25747 

.42347 

.26514 

.43596 

.27288 

.44822 

.28068 

£0  60 

9.41094 

0.25760 

9.42368 

0.26526 

9.43617 

0.27300 

9.44842 

0.28081 

266 


Table  10.    Haversine  Table 


s   ' 

4*  .7£m    64° 

4*  20OT    65° 

4h  24m   66° 

4*  28"1    67° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.44842 

0.28081 

9.46043 

0.28869 

9.47222 

0.29663 

9.48378 

0.30463 

4  l 

.44862 

.28095 

.46063 

.28882 

.47241 

.29676 

.48397 

.30477 

8   2 

.44882 

.28108 

.46083 

.28895 

.47261 

.29690 

.48416 

.30490 

12   3 

.44903 

.28121 

.46103 

.28909 

.47280 

.29703 

.48435 

.30504 

16   4 

.44923 

.28134 

.46123 

.28922 

.47300 

.29716 

.48454 

.30517 

20   5 

9.44943 

0.28147 

9.46142 

0.28935 

9.47319 

0.29730 

9.48473 

0.30530 

24   6 

.44963 

.28160 

.46162 

.28948 

.47338 

.29743 

.48492 

.30544 

28   7 

.44983 

.28173 

.46182 

.28961 

.47358 

.29756 

.48511 

.30557 

32   8 

.45003 

.28186 

.46202 

.28975 

.47377 

.29770 

.48530 

.30571 

36   9 

.45024 

.28199 

.46222 

.28988 

.47397 

.29783 

.48549 

.30584 

40  10 

9.45044 

0.28212 

9.46241 

0.29001 

9.47416 

0.29796 

9.48568 

0.30597 

44  11 

.45064 

.28225 

.46261 

.29014 

.47435 

.29809 

.48587 

.30611 

48  12 

.45084 

.28238 

.46281 

.29027 

.47455 

.29823 

.48607 

.30624 

52  13 

.45104 

.28252 

.46301 

.29041 

.47474 

.29836 

.48626 

.30638 

56  14 

.45124 

.28265 

.46320 

.29054 

.47493 

.29849 

.48645 

.30651 

s   ' 

4h  17m     64o 

4h  21m    65° 

4h  25m    66° 

4h  29"1    67° 

0  15 

9.45144 

0.28278 

9.46340 

0.29067 

9.47513 

0.29863 

9.48664 

0.30664 

4  16 

.45165 

.28291 

.46360 

.29080 

.47532 

.29876 

.48683 

.30678 

5  17 

.45185 

.28304 

.46380 

.29093 

.47552 

.29889 

.48702 

.30691 

12  18 

.45205 

.28317 

.46399 

.29107 

.47571 

.29903 

.48720 

.30705 

J0  19 

.45225 

.28330 

.46419 

.29120 

.47590 

.29916 

.48739 

.30718 

20  20 

9.45245 

0.28343 

9.46439 

0.29133 

9.47610 

0.29929 

9.48758 

0.30732 

24  21 

.45265 

.28356 

.46458 

.29146 

.47629 

.29943 

.48777 

.30745 

25  22 

.45285 

.28369 

.46478 

.29160 

.47648 

.29956 

.48796 

.30758 

32  23 

.45305 

.28383 

.46498 

.29173 

.47668 

.29969 

.48815 

.30772 

3£  24 

.45325 

.28396 

.46517 

.29186 

.47687 

.29983 

.48834 

.30785 

40  25 

9.45345 

0.28409 

9.46537 

0.29199 

9.47706 

0.29996 

9.48853 

0.30799 

44  26 

.45365 

.28422 

.46557 

.29212 

.47725 

.30009 

.48872 

.30812 

45  27 

.45385 

.28435 

.46576 

.29226 

.47745 

.30023 

.48891 

.30826 

52  28 

.45405 

.28448 

.46596 

.29239 

.47764 

.30036 

.48910 

.30839 

5(5  29 

.45426 

.28461 

.46616 

.29252 

.47783 

.30049 

.48929 

.30852 

s   ' 

4h  18m    64° 

4^  22m    65° 

4h  26m    66° 

4h  30m     67° 

0  30 

9.45446 

0.28474 

9.46635 

0.29265 

9.47803 

0.30063 

9.48948 

0.30866 

4  31 

.45466 

.28488 

.46655 

.29279 

.47822 

.30076 

.48967 

.30879 

8  32 

.45486 

.28501 

.46675 

.29292 

.47841 

.30089 

.48986 

.30893 

.72  33 

.45506 

.28514 

.46694 

.29305 

.47860 

.30103 

.49004 

.30906 

.?£  34 

.45526 

.28527 

.46714 

.29318 

.47880 

.30116 

.49023 

.30920 

20  35 

9.45546 

0.28540 

9.46733 

0.29332 

9.47899 

0.30129 

9.49042 

0.30933 

24  36 

.45566 

.28553 

.46753 

.29345 

.47918 

.30143 

.49061 

.30946 

28  37 

.45586 

.28566 

.46773 

.29358 

.47937 

.30156 

•49080 

.30960 

32  38 

.45606 

.28580 

.46792 

.29371 

.47957 

.30169 

.49099 

.30973 

35  39 

.45625 

.28593 

.46812 

.29385 

.47976 

.30183 

.49118 

.30987 

40  40 

9.45645 

0.28606 

9.46831 

0.29398 

9.47995 

0.30196 

9.49137 

0.31000 

44  41 

.45665 

.28619 

.46851 

.29411 

.48014 

.30209 

.49155 

.31014 

48  42 

.45685 

.28632 

.46871 

.29424 

.48033 

.30223 

.49174 

.31027 

52  43 

.45705 

.28645 

.46890 

.29438 

.48053 

.30236 

.49193 

.31041 

5£  44 

.45725 

.28658 

.46910 

.29451 

.48072 

.30249 

.49212 

.31054 

s   ' 

4^  19m    64° 

4h  23™    65° 

jrh  27m    66° 

4h  sim    67° 

0  45 

9.45745 

0.28672 

9.46929 

0.29464 

9.48091 

0.30263 

9.49231 

0.31068 

4  46 

.45765 

.28685 

.46949 

.29477 

.48110 

.30276 

.49250 

.31081 

5  47 

.45785 

.28698 

.46968 

.29491 

.48129 

.30290 

.49268 

.31095 

^2  48 

.45805 

.28711 

.46988 

.29504 

.48148 

.30303 

.49287 

.31108 

16  49 

.45825 

.28724 

.47007 

.29517 

.48168 

.30316 

.49306 

.31121 

20  50 

9.45845 

0.28737 

9.47027 

0.29530 

9.48187 

0.30330 

9.49325 

0.31135 

24  51 

.45865 

.28751 

.47046 

.29544 

.48206 

.30343 

.49344 

.31148 

28  52 

.45884 

.28764 

.47066 

.29557 

.48225 

.30356 

.49362 

.31162 

32  53 

.45904 

.28777 

.47085 

.29570 

.48244 

.30370 

.49481 

.31175 

36  54 

.45924 

.28790 

.47105 

.29583 

.48263 

.30383 

.49400 

.31189 

40  55 

9.45944 

0.28803 

9.47124 

0.29597 

9.48282 

0.30397 

9.49419 

0.31202 

44  56 

.45964 

.28816 

.47144 

.29610 

.48302 

.30410 

.49437 

.31216 

48  57 

.45984 

.28830 

.47163 

.29623 

.48321 

.30423 

.49456 

.31229 

52  58 

.46004 

.28843 

,47183 

.29637 

.48340 

.30437 

.49475 

.31243 

66  59 

.46023 

.28856 

.47202 

.29650 

.48359 

.30450 

.49494 

.31256 

60  60 

9.46043 

0.28869 

9.47222 

0.29663 

9.48378 

0.30463 

9.49512 

0.31270 

Table  10.    Haversine  Table 


26< 


s   ' 

4*  32™    68° 

4*  36™    69° 

4h  40™   70° 

4*  44m   71° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.49512 

0.31270 

9.50626 

0.32082 

9.51718 

0.32899 

9.52791 

0.33722 

4   1 

.49531 

.31283 

.50644 

.32095 

.51736 

.32913 

.52809 

.33735 

8   2 

.49550 

.31297 

.50662 

.32109 

.51754 

.32926 

.52826 

.33749 

n   3 

.49568 

.31310 

.50681 

.32122 

.51772 

.32940 

.52844 

.33763 

16   4 

.49587 

.31324 

.50699 

.32136 

.51790 

.32954 

.52862 

.33777 

20   5 

9.49606 

0.31337 

9.50717 

0.32150 

9.51808 

0.32967 

9.52879 

0.33790 

24   6 

.49625 

.31351 

.50736 

.32163 

.51826 

.32981 

.52897 

.33804 

28   7 

.49643 

.31364 

.50754 

.32177 

.51844 

.32995 

.52915 

.33818 

32   8 

.49662 

.31378 

.50772 

.32190 

.51862 

.33008 

.52932 

.33832 

36   9 

.49681 

.31391 

.50791 

.32204 

.51880 

.33022 

.52950 

.33845 

40  10 

9.49699 

0.31405 

9.50809 

0.32217 

9.51898 

0.33036 

9.52968 

0.33859 

44  11 

.49718 

.31418 

.50827 

.32231 

.51916 

.33049 

.52985 

.33873 

48  12 

.49737 

.31432 

.50846 

.32245 

.51934 

.33063 

.53003 

.33887 

52  13 

.49755 

.31445 

.50864 

.32258 

.51952 

.33077 

.53021 

.33900 

56  14 

.49774 

.31459 

.50882 

.32272 

.51970 

.33090 

.53038 

.33914 

s   ' 

4*  S3"1    68° 

4*  37m    69° 

4^  4im   70° 

4h  45m    71° 

0  15 

9.49793 

0.31472 

9.50901 

0.32285 

9.51988 

0.33104 

9.53056 

0.33928 

4  16 

.49811 

.31486 

.50919 

.32299 

.52006 

.33118 

.53073 

.33942 

S  17 

.49830 

.31499 

.50937 

.32313 

.52024 

.33132 

.53091 

.33956 

1  >   18 

.49849 

.31513 

.50956 

.32326 

.52042 

.33145 

.53109 

.33969 

/0  19 

.49867 

.31526 

.50974 

.32340 

.52060 

.33159 

.53126 

.33983 

20  20 

9.49886 

0.31540 

9.50992 

0.32353 

9.52078 

0.33173 

9.53144 

0.33997 

24  21 

.49904 

.31553 

.51010 

.32367 

.52096 

.33186 

.53162 

.34011 

2S  22 

.49923 

.31567 

.51029 

.32381 

.52114 

.33200 

.53179 

.34024 

32  23 

.49942 

.31580 

.51047 

.32394 

.52132 

.33214 

.53197 

.34038 

35  24 

.49960 

.31594 

.51065 

.32408 

.52150 

.33227 

.53214 

.34052 

40  25 

9.49979 

0.31607 

9.51083 

0.32422 

9.52168 

0.33241 

9.53232 

0.34066 

44  26 

.49997 

.31621 

.51102 

.32435 

.52185 

.33255 

.53249 

.34080 

45  27 

.50016 

.31634 

.51120 

.32449 

.52203 

.33269 

.53267 

.34093 

.52  28 

.50034 

.31648 

.51138 

.32462 

.52221 

.33282 

.53285 

.34107 

5(7  29 

.50053 

.31661 

.51156 

.32476 

.52239 

.33296 

.53302 

.34121 

s   ' 

4h  34m    68° 

4*  38m    69° 

4h  4%™   70° 

4*  46™   71° 

0  30 

9.50072 

0.31675 

9.51174 

0.32490 

9.52257 

0.33310 

9.53320 

0.34135 

4  31 

.50090 

.31688 

.51193 

.32503 

.52275 

.33323 

.53337 

.34149 

8  32 

.50109 

.31702 

.51211 

.32517 

.52293 

.33337 

.53355 

.34162 

J2  33 

.50127 

.31716 

.51229 

.32531 

.52311 

.33351 

.53372 

.34176 

16  34 

.50146 

.31729 

.51247 

.32544 

.52328 

.33365 

.53390 

.34190 

20  35  9.50164 

0.31742 

9.51265 

0.32558 

9.52346 

0.33378 

9.53407 

0.34204 

24  36 

.50183 

.31756 

.51284 

.32571 

.52364 

.33392 

.53425 

.34218 

25  37 

.50201 

.31770 

.51302 

.32585 

.52382 

.33406 

.53442 

.34231 

32  38 

.50220 

.31783 

.51320 

.32599 

.52400 

.33419 

.53460 

.34245 

36  39 

.50238 

.31797 

.51338 

.32612 

.52418 

.33433 

.53477 

.34259 

40  40 

9.50257 

0.31810 

9.51356 

0.32626 

9.52436 

0.33447 

9.53495 

0.34273 

44  41 

.50275 

.31824 

.51374 

.32640 

.52453 

.33461 

.53512 

.34287 

48  42 

.50294 

.31837 

.51393 

.32653 

.52471 

.33474 

.53530 

.34300 

52  43 

.50312 

.31851 

.51411 

.32667 

.52489 

.33488 

.53547 

.34314 

56  44 

.50331 

.31865 

.51429 

.32681 

.52507 

.33502 

.53565 

.34328 

s   ' 

4*  35m    68° 

4*  39m    69° 

4h  ^^m   70o 

4h  47m     71° 

0  45 

9.50349 

0.31878 

9.51447 

0.32694 

9.52525 

0.33515 

9.53582 

0.34342 

4  46 

.50368 

.31892 

.51465 

.32708 

.52542 

.33529 

.53600 

.34356 

2  47 

.50386 

.31905 

.51483 

.32721 

.52560 

.33543 

.53617 

.34369 

12  48 

.50405 

.31919 

.51501 

.32735 

.52578 

.33557 

.53635 

.34383 

/0  49 

.50423 

.31932 

.51519 

.32749 

.52596 

.33570 

.53652 

.34397 

20  50 

9.50442 

0.31946 

9.51538 

0.32762 

9.52613 

0.33584 

9.53670 

0.34411 

24  51 

.50460 

.31959 

.51556 

.32776 

.52631 

.33598 

.53687 

.34425 

28  52 

.50478 

.31973 

.51574 

.32790 

.52649 

.33612 

.53704 

.34439 

32  53 

.50497 

.31987 

.51592 

.32803 

.52667 

.33625 

.53722 

.34452 

36  54 

.50515 

.32000 

.51610 

.32817 

.52684 

.33639 

.53739 

.34466 

40  55 

9.50534 

0.32014 

9.51628 

0.32831 

9.52702 

0.33653 

9.53757 

0.34480 

44  56 

.50552 

.32027 

.51646 

.32844 

.52720 

.33667 

.53774 

.34494 

48  57 

.50570 

.32041 

.51664 

.32858 

.52738 

.33680 

.53792 

.34508 

52  58 

.50589 

.32054 

.51682 

.32872 

.52755 

.33694 

.53809 

.34521 

56  59 

.50607 

.32068 

.51700 

.32885 

.52773 

.33708 

.53826 

.34535 

60  60 

9.50626 

0.32082 

9.51718 

0.32899 

9.52791 

0.33722 

9.53844 

0.34549 

268 


Table  10.    Haversine  Table 


s   ' 

4h  48m    72° 

4*  52m    73° 

^h  5ffm     74° 

5h  <yn    75° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.53844 

0.34549 

9.54878 

0.35381 

9.55893 

0.36218 

9.56889 

0.37059 

4   1 

.53861 

.34563 

.54895 

.35395 

.55909 

.36232 

.56906 

.37073 

8   2 

.53879 

.34577 

.54912 

.35409 

.55926 

.36246 

.56922 

.37087 

12   3 

.53896 

.34591 

.54929 

.35423 

.55943 

.36260 

.56939 

.37101 

16   4 

.53913 

.34604 

.54946 

.35437 

.55960 

.36274 

.56955 

.37115 

20   5 

9.53931 

0.34618 

9.54963 

0.35451 

9.55976 

0.36288 

9.56972 

0.37129 

24   6 

.53948 

.34632 

.54980 

.35465 

.55993 

.36302 

.56988 

.37143 

28   1 

.53966 

.34646 

.54997 

.35479 

.56010 

.36316 

.57005 

.37157 

32   8 

.53983 

.34660 

.55014 

.35493 

.56027 

.36330 

.57021 

.37171 

36   9 

.54000 

.34674 

.55031 

.35507 

.56043 

.36344 

.57037 

.37186 

40  10 

9.54017 

0.34688 

9.55048 

0.35521 

9.56060 

0.36358 

9.57054 

0.37200 

44  11 

.54035 

.34701 

.55065 

.35534 

.56077 

.36372 

.57070 

.37214 

48  12 

.54052 

.34715 

.55082 

.35548 

.56093 

.36386 

.57087 

.37228 

52  13 

.54069 

.34729 

.55099 

.35562 

.56110 

.36400 

.57103 

.37242 

56  14 

.54087 

.34743 

.55116 

.35576 

.56127 

.36414 

.57119 

.37256 

s   ' 

4h  J^Q™    72° 

4h  53™    73° 

4h  67m     74° 

Qh  im      750 

0  15 

9.54104 

0.34757 

9.55133 

0.35590 

9.56144 

0.36428 

9.57136 

0.37270 

4  16 

.54121 

.34771 

.55150 

.35604 

.56160 

.36442 

.57152 

.37284 

5  17 

.54139 

.34784 

.55167 

.35618 

.56177 

.36456 

.57169 

.37298 

12  18 

.54156 

.34798 

.55184 

.35632 

.56194 

.36470 

.57185 

.37312 

^6?  19 

.54173 

.34812 

.55201 

.35646 

.56210 

.36484 

.57201 

.37326 

20  20 

9.54190 

0.34826 

9.55218 

0.35660 

9.56227 

0.36498 

9.57218 

0.37340 

24  21 

.54208 

.34840 

.55235 

.35674 

.56244 

.36512 

.57234 

.37354 

25  22 

.54225 

.34854 

.55252 

.35688 

.56260 

.36526 

.57250 

.37368 

32  23 

.54242 

.34868 

.55269 

.35702 

.56277 

.36540 

.57267 

.37382 

35  24 

.54260 

.34882 

.55286 

.35716 

.56294 

.36554 

.57283 

.37397 

40  25 

9.54277 

0.34895 

9.55303 

0.35730 

9.56310 

0.36568 

9.57299 

0.37411 

44  26 

.54294 

.34909 

.55320 

.35743 

.56327 

.36582 

.57316 

.37425 

48  27 

.54311 

.34923 

.55337 

.35757 

.56343 

.36596 

.57332 

.37439 

52  28 

.54329 

.34937 

.55354 

.35771 

.56360 

.36610 

.57348 

.37453 

56?  29 

.54346 

.34951 

.55370 

.35785 

.56377 

.36624 

.57365 

.37467 

s   ' 

4h  5Qm    72° 

4h  54m    73° 

jji  5gm    74° 

5h  2m    75° 

0  30 

9.54363 

0.34965 

9.55387 

0.35799 

9.56393 

0.36638 

9.57381 

0.37481 

4  31 

.54380 

.34979 

.55404 

.35813 

.56410 

.36652 

.57397 

.37495 

8  32 

.54397 

.34992 

.55421 

.35827 

.56426 

.36666 

.57414 

.37509 

^2  33 

.54415 

.35006 

.55438 

.35841 

.56443 

.36680 

.57430 

.37523 

.76?  34 

.54432 

.35020 

.55455 

.35855 

.56460 

.36694 

.57446 

.37537 

20  35 

9.54449 

0.35034 

9.55472 

0.35869 

9.56476 

0.36708 

9.57463 

0.37551 

24  36 

.54466 

.35048 

.55489 

.35883 

.56493 

.36722 

.57479 

.37566 

28  37 

.54483 

.35062 

.55506 

.35897 

.56509 

.36736 

.57495 

.37580 

32  38 

.54501 

.35076 

.55523 

.35911 

.56526 

.36750 

.57511 

.37594 

36?  39 

.54518 

.35090 

.55539 

.35925 

.56543 

.36764 

.57528 

.37608 

40  40 

9.54535 

0.35103 

9.55556 

0.35939 

9.56559 

0.36778 

9.57544 

0.37622 

44  41 

.54552 

.35117 

.55573 

.35953 

.56576 

.36792 

.57560 

.37636 

48  42 

.54569 

.35131 

.55590 

.35967 

.56592 

.36806 

.57577 

.37650 

52  43 

.54587 

.35145 

.55607 

.35981 

.56609 

.36820 

.57593 

.37664 

56?  44 

.54604 

.35159 

.55624 

.35995 

.56625 

.36834 

.57609 

.37678 

s   ' 

4h  51^    72° 

4*  55m    73° 

4h  59m    74° 

5h  3™    75° 

0  45 

9.54621 

0.35173 

9.55641 

0.36009 

9.56642 

0.36848 

9.57625 

0.37692 

4  46 

.54638 

.35187 

.55657 

.36023 

.56658 

.36862 

.57642 

.37706 

5  47 

.54655 

.35201 

.55674 

.36036 

.56675 

.36877 

.57658 

.37721 

.72  48 

.54672 

.35215 

.55691 

.36050 

.56692 

.36891 

.57674 

.37735 

16  49 

.54689 

.35228 

.55708 

.36064 

.56708 

.36905 

.57690 

.37749 

26>  50 

9.54707 

0.35242 

9.55725 

0.36078 

9.56725 

0.36919 

9.57706 

0.37763 

24  51 

.54724 

.35256 

.55742 

.36092 

.56741 

.36933 

.57723 

.37777 

28  52 

.54741 

.35270 

.55758 

.36106 

.56758 

.36947 

.57739 

.37791 

32  53 

.54758 

.35284 

.55775 

.36120 

.56774 

.36961 

.57755 

.37805 

36?  54 

.54775 

.35298 

.55792 

.36134 

.56791 

.36975 

.57771 

.37819 

40  55 

9.54792 

0.35312 

9.55809 

0.36148 

9.56807 

0.36989 

9.57787 

0.37833 

44  56 

.54809 

.35326 

.55826 

.36162 

.56824 

.37003 

.57804 

.37847 

48  57 

.54826 

.35340 

.55842 

.36176 

.56840 

.37017 

.57820 

.37862 

52  58 

.54843 

.35354 

.55859 

.36190 

.56856 

37031 

.57836 

.37876 

56?  59 

.54860 

.35368 

.55876 

.36204 

.56873 

.37045 

.57852 

.37890 

6?6>  60 

9.54878 

0.35381 

9.55893 

0.36218 

9.56889 

0.37059 

9.57868 

0.37904 

Table  10.    Haversine  Table 


269 


s    ' 

5*  4m     76° 

5h  gm      77° 

5h  i2m    78° 

5h  16m    79° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.57868 

0.37904 

9.58830 

0.38752 

9.59774 

0.39604 

9.60702 

0.40460 

4   1 

.57885 

.37918 

.58846 

.38767 

.59790 

.39619 

.60717 

.40474 

8   2 

.57901 

.37932 

.58862 

.38781 

.59806 

.39633 

.60733 

.40488 

12   3 

.57917 

.37946 

.58878 

.38795 

.59821 

.39647 

.60748 

.40502 

16   4 

.57933 

.37960 

.58893 

.38809 

.59837 

.39661 

.60763 

.40517 

20   5 

9.57949 

0.37974 

9.58909 

0.38823 

9.59852 

0.39676 

9.60779 

0.40531 

24   6 

.57965 

.37989 

.58925 

.38837 

.59868 

.39690 

.60794 

.40545 

28   7 

.57981 

.38003 

.58941 

.38852 

.50883 

.39704 

.60809 

.40560 

32   8 

.57998 

.38017 

.58957 

.38866 

.59899 

.39718 

.60825 

.40574 

36   9 

.58014 

.38031 

.58973 

.38880 

.59915 

.39732 

.60840 

.40588 

40  10 

9.58030 

0.38045 

9.58989 

0.38894 

9.59930 

0.39746 

9.60855 

0.40602 

44  11 

.58046 

.38059 

.59004 

.38908 

.59946 

.39761 

.60870 

.40617 

48  12 

.58062 

.38073 

.59020 

.38923 

.59961 

.39775 

.60886 

.40631 

52  13 

.58078 

.38087 

.59036 

.38937 

.59977 

.39789 

.60901 

.40645 

56  14 

.58094 

.38102 

.59052 

.38951 

.59992 

.39803 

.60916 

.40660 

s    ' 

5h  5m     76° 

6h  9m      77° 

5h  IS"1    78° 

6h  17m     79° 

0  15 

9.58110 

0.38116 

9.59068 

0.38965 

9.60008 

0.39818 

9.60931 

0.40674 

4  16 

.58126 

.38130 

.59083 

.38979 

.60023 

.39832 

.60947 

.40688 

S  17 

.58143 

.38144 

.59099 

.38994 

.60039 

.39846 

.60962 

.40702 

12  18 

.58159 

.38158 

.59115 

.39008 

.60054 

.39861 

.60977 

.40717 

16  19 

.58175 

.38172 

.59131 

.39022 

.60070 

.39875 

.60992 

.40731 

20  20 

9.58191 

0.38186 

9.59147 

0.39036 

9.60085 

0.39889 

9.61008 

0.40745 

24  21 

.58207 

.38200 

.59162 

.39050 

.60101 

.39903 

.61023 

.40760 

2S  22 

.58223 

.38215 

.59178 

.39064 

.60116 

.39918 

.61038 

.40774 

32  23 

.58239 

.38229 

.59194 

.39079 

.60132 

.39932 

.61053 

.40788 

35  24 

.58255 

.38243 

.59210 

.39093 

.60147 

.39946 

.61069 

.40802 

40  25 

9.58271 

0.38257 

9.59225 

0.39107 

9.60163 

0.39960 

9.61084 

0.40817 

44  26 

.58287 

.38271 

.59241 

.39121 

.60178 

.39975 

.61099 

.40831 

45  27 

.58303 

.38285 

.59257 

.39135 

.60194 

.39989 

.61114 

.40845 

52  28 

.58319 

.38299 

.59273 

.39150 

.60209 

.40003 

.61129 

.40860 

55  29 

.58335 

.38314 

.59289 

.39164 

.60225 

.40017 

.61145 

.40874 

s    ' 

5h  &n      76° 

5h  10m    77° 

5h  14™    78° 

5h  is™    79° 

0  30 

9.58351 

0.38328 

9.59304 

0.39178 

9.60240 

0.40032 

9.61160  10.40888 

4  31 

.58367 

.38342 

.59320 

.39192 

.60256 

.40046 

.61175 

.40903 

8  32 

.58383 

.38356 

.59336 

.39206 

.60271 

.40060 

.61190 

.40917 

12  33 

.58399 

.38370 

.59351 

.39221 

.60287 

.40074 

.61205 

.40931 

16  34 

.58415 

.38384 

.59367 

.39235 

.60302 

.40089 

.61221 

.40945 

20  35 

9.58431 

0.38398 

9.59383 

0.39249 

9.60318 

0.40103 

9.61236 

0.40960 

24  36 

.58447 

.38413 

.59399 

.39263 

.60333 

.40117 

.61251 

.40974 

28  37 

.58463 

.38427 

.59414 

.39277 

.60348 

.40131 

.61266 

.40988 

32  38 

.58479 

.38441 

.59430 

.39292 

.60364 

.40146 

.61281 

.41003 

35  39 

.58495 

.38455 

.59446 

.39306 

.60379 

.40160 

.61296 

.41017 

40  40 

9.58511 

0.38469 

9.59461 

0.39320 

9.60395 

0.40174 

9.61312 

0.41031 

44  41 

.58527 

.38483 

.59477 

.39334 

.60410 

.40188 

.61327 

.41046 

48  42 

.58543 

.38498 

.59493 

.39348 

.60426 

.40203 

.61342 

.41060 

52  43 

.58559 

.38512 

.59508 

.39363 

.60441 

.40217 

.61357 

.41074 

56  44 

.58575 

.38526 

.59524 

.39377 

.60456 

.40231 

.61372 

.41089 

s    ' 

5h  7"»       76° 

5h  nm     77° 

5h  15>n     ?8o 

6h  igm    79° 

0  45 

9.58591 

0.38540 

9.59540 

0.39391 

9.60472 

0.40245 

9.61387 

0.41103 

4  46 

.58607 

.38554 

.59556 

.39405 

.60487 

.40260 

.61402 

.41117 

5  47 

.58623 

.38568 

.59571 

.39420 

.60502 

.40274 

.61417 

.41131 

12  48 

.58639 

.38582 

.59587 

.39434 

.60518 

.40288 

.61433 

.41146 

16  49 

.58655 

.38597 

.59602 

.39448 

.60533 

.40303 

.61448 

.41160 

20  50 

9.58671 

0.38611 

9.59618 

0.39462 

9.60549 

0.40317 

9.61463 

0.41174 

24  51 

.58687 

.38625 

.59634 

.39476 

.60564 

.40331 

.61478 

.41189 

28  52 

.58703 

.38639 

.59649 

.39491 

.60579 

.40345 

.61493 

.41203 

32  53 

.58719 

.38653 

.59665 

.39505 

.60595 

.40360 

.61508 

.41217 

36  54 

.58735 

.38667 

.59681 

.39519 

.60610 

.40374 

.61523 

.41232 

40  55 

9.58750 

0.38682 

9.59696 

0.39533 

9.60625 

0.40388 

9.61538 

0.41246 

44  56 

.58766 

.OOOJD 

.59712 

.39548 

.60641 

.40402 

.61553 

.41260 

48  57 

.58782 

.38710 

.59728 

.39562 

.60656 

.40417 

.61568 

.41275 

52  58 

.58798 

.38724 

.59743 

.39576 

.60671 

.40431 

.61583 

.41289 

56  59 

.58814 

.38738 

.59759 

.39590 

.60687 

.40445 

.61598 

.41303 

60  60 

9.58830 

0.38752 

9.59774 

0.39604 

9.60702 

0.40460 

9.61614 

0.41318 

270 


Table  10.    Haversine  Table 


s   ' 

5*  20m    80° 

5h  24m    81° 

gh  28m    82° 

•  oh  32m    83° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.61614 

0.41318 

9.62509 

0.42178 

9.63389 

0.43041 

9.64253 

0.43907 

4  1 

.61629 

.41332 

.62524 

.42193 

.63403 

.43056 

.64267 

.43921 

8   2 

.61644 

.41346 

.62538 

.42207 

.63418 

.43070 

.64281 

.43935 

12   3 

.61659 

.41361 

.62553 

.42221 

.63432 

.43085 

.64296 

.43950 

16   4 

.61674 

.41375 

.62568 

.42236 

.63447 

.43099 

.64310 

.43964 

20   5 

9.61689 

0.41389 

9.62583 

0.42250 

9.63461 

0.43113 

0.64324 

0.43979 

24   6 

.61704 

.41404 

.62598 

.42264 

.63476 

.43128 

.64339 

.43993 

28   7 

.61719 

.41418 

.62612 

.42279 

.63490 

.43142 

.64353 

.44008 

32   8 

.61734 

.41432 

.62627 

.42293 

.63505 

.43157 

.64367 

.44022 

36   9 

.61749 

.41447 

.62642 

.42308 

.63519 

.43171 

.64381 

.44036 

40  10 

9.61764 

0.41461 

9.62657 

0.42322 

9.63534 

0.43185 

9.64396 

0.44051 

44  11 

.61779 

.41475 

.62671 

.42336 

.63548 

.43200 

.64410 

.44065 

48  12 

.61794 

.41490 

.62686 

.42351 

.63563 

.43214 

.64424 

.44080 

52  13 

.61809 

.41504 

.62701 

.42365 

.63577 

.43229 

.64438 

.44094 

56  14 

.61824 

.41518 

.62716 

.42379 

.63592 

.43243 

.64452 

.44109 

s    ' 

gh  21m   80° 

5h  25™    81° 

gh  29m    82° 

gh  33m    83° 

0  15 

9.61839 

0.41533 

9.62730 

0.42394 

9.63606 

0.43257 

9.64467 

0.44123 

4  16 

.61854 

.41547 

.62745 

.42408 

.63621 

.43272 

.64481 

.44138 

5  17 

.61869 

.41561 

.62760 

.42423 

.63635 

.43286 

.64495 

.44152 

12  18 

.61884 

.41576 

.62774 

.42437 

.63649 

.43301 

.64509 

.44166 

15  19 

.61899 

.41590 

.62789 

.42451 

.63664 

.43315 

.64523 

.44181 

£0  20 

9.61914 

0.41604 

9.62804 

0.42466 

9.63678 

0.43330 

9.64538 

0.44195 

24  21 

.61929 

.41619 

.62819 

.42480 

.63693 

.43344 

.64552 

.44210 

^o  22 

.61944 

.41633 

.62833 

.42494 

.63707 

.43358 

.64566 

.44224 

32  23 

.61959 

.41647 

.62848 

.42509 

.63722 

.43373 

.64580 

.44239 

35  24 

.61974 

.41662 

.62863 

.42523 

.63736 

.43387 

.64594 

.44253 

40  25 

9.61989 

0.41676 

9.62877 

0.42538 

9.63751 

0.43402 

9.64609 

0.44268 

44  26 

.62003 

.41690 

.62892 

.42552 

.63765 

.43416 

.64623 

.44282 

45  27 

.62018 

.41705 

.62907 

.42566 

.63779 

.43430 

.64637 

.44296 

52  28 

.62033 

.41719 

.62921 

.42581 

.63794 

.43445 

.64651 

.44311 

55  29 

.62048 

.41733 

.62936 

.42595 

.63808 

.43459 

.64665 

.44325 

s    ' 

5h  2%m    80° 

5h  25m    81° 

gh  som    82° 

gh  34m    83° 

0  30 

9.62063 

0.41748 

9.62951 

0.42610 

9.63823 

0.43474 

9.64679 

0.44340 

4  31 

.62078 

.41762 

.62965 

.42624 

.63837 

.43488 

.64694 

.44354 

8  32 

.62093 

.41776 

.62980 

.42638 

.63851 

.43503 

.64708 

.44369 

^2  33 

.62108 

.41791 

.62995 

.42653 

.63866 

.43517 

.64722 

.44383 

^5  34 

.62123 

.41805 

.63009 

.42667 

.63880 

.43531 

.64736 

.44398 

20  35 

9.62138 

0.41819 

9.63024 

0.42681 

9.63895 

0.43546 

9.64750 

0.44412 

#4  36 

.62153 

.41834 

.63039 

.42696 

.63909 

.43560 

.64764 

.44427 

28  37 

.62168 

.41848 

.63063 

.42710 

.63923 

.43575 

.64778 

.44441 

32  38 

.62182 

.41862 

.63068 

.42725 

.63938 

.43589 

.64793 

.44455 

35  39 

.62197 

.41877 

.63082 

.42739 

.63952 

.43603 

.64807 

.44470 

40  40 

9.62212 

0.41891 

9.63097 

0.42753 

9.63966 

0.43618 

9.64821 

0.44484 

44  41 

.62227 

.41905 

.63112 

.42768 

.63981 

.43632 

.64835 

.44499 

48  42 

.62242 

.41920 

.63126 

.42782 

.63995 

.43647 

.64849 

.44513 

52  43 

.62257 

.41934 

.63141 

.42797 

.64010 

.43661 

.64863 

.44528 

55  44 

.62272 

.41949 

.63156 

.42811 

.64024 

.43676 

.64877 

.44542 

s    ' 

5h  23m    80° 

gh  27m    81° 

gh  31m    82° 

gh  35m    83° 

0  45 

9.62287 

0.41963 

9.63170 

0.42825 

9.64038 

0.43690 

9.64891 

0.44557 

4  46 

.62301 

.41977 

.63185 

.42840 

.64053 

.43704 

.64905 

.44571 

5  47 

.62316 

.41992 

.63199 

.42854 

.64067 

.43719 

.64919 

.44586 

.72  48 

.62331 

.42006 

.63214 

.42869 

.64081 

.43733 

.64934 

.44600 

16  49 

.62346 

.42020 

.63228 

.42883 

.64096 

.43748 

.64948 

.44614 

20  50 

9.62361 

0.42035 

9.63243 

0.42897 

9.64110 

0.43762 

9.64962 

0.44629 

24  51 

.62376 

.42049 

.63258 

.42912 

.64124 

.43777 

.64976 

.44643 

28  52 

.62390 

.42063 

.63272 

.42926 

.64139 

.43791 

.64990 

.44658 

32  53 

.62405 

.42078 

.63287 

.42941 

.64153 

.43805 

.65004 

.44672 

35  54 

.62420 

.42092 

.63301 

.42955 

.64167 

.43820 

.65018 

.44687 

40  55 

9.62435 

0.42106 

9.63316 

0.42969 

9.64181 

0.43834 

9.65032 

0.44701 

44  56 

.62450 

.42121 

.63330 

.42984 

.64196 

.43849 

.65046 

.44716 

48  57 

.62464 

.42135 

.63345 

.42998 

.64210 

.43863 

.65060 

.44730 

52  58 

.62479 

.42150 

.63360 

.43013 

.64224 

.43878 

.65074 

.44745 

55  59 

.62494 

.42164 

.63374 

.43027 

.64239 

.43892 

.65088 

.44759 

60  60 

9.62509 

0.42178 

9.63389 

0.43041 

9.64253 

0.43907 

9.65102 

0.44774 

Table  10.    Haversine  Table 


271 


s    ' 

5*  36m    84° 

5*  40m    85° 

5*  44m    86° 

5h  48m    87° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0   0 

9.65102 

0.44774 

9.65937 

0.45642 

9.66757 

0.46512 

9.67562 

0.47383 

4  l 

.65116 

.44788 

.65950 

.45657 

.66770 

.46527 

.67576 

.47398 

8   2 

.65130 

.44803 

.65964 

.45671 

.66784 

.46541 

.67589 

.47412 

12   3 

.65144 

.44817 

.65978 

.45686 

.66797 

.46556 

.67602 

.47427 

16   4 

.65158 

.44831 

.65992 

.45700 

.66811 

.46570 

.67616 

.47441 

20   5 

9.65172 

0.44846 

9.66006 

0.45715 

9.66824 

0.46585 

9.67629 

0.47456 

24   6 

.65186 

.44860 

.66019 

.45729 

.66838 

.46599 

.67642 

.47470 

28   7 

.65200 

.44875 

.66033 

.45744 

.66851 

.46614 

.67656 

.47485 

32   8 

.65214 

.44889 

.66047 

.45758 

.66865 

.46628 

.67669 

.47499 

36   9 

.65228 

.44904 

.66061 

.45773 

.66878 

.46643 

.67682 

.47514 

40  10 

9.65242 

0.44918 

9.66074 

0.45787 

9.66892 

0.46657 

9.67695 

0.47528 

44  11 

.65256 

.44933 

.66088 

.45802 

.66905 

.46672 

.67709 

.47543 

48  12 

.65270 

.44947 

.66102 

.45816 

.66919 

.46686 

.67722 

.47558 

52  13 

.65284 

.44962 

.66116 

.45831 

.66932 

.46701 

.67735 

.47572 

56  14 

.65298 

.44976 

.66129 

.45845 

.66946 

.46715 

.67748 

.47587 

s   ' 

oh  37m    84° 

5h  41m    85° 

5^  45m    86° 

5h  49m    87° 

0  15 

9.65312 

0.44991 

9.66143 

0.45860 

9.66959 

0.46730 

9.67762 

0.47601 

4  16 

.65326 

.45005 

.66157 

.45874 

.66973 

.46744 

.67775 

.47616 

5  17 

.65340 

.45020 

.66170 

.45889 

.66986 

.46759 

.67788 

.47630 

12  18 

.65354 

.45034 

.66184 

.45903 

.67000 

.46773 

.67801 

.47645 

76  19 

.65368 

.45048 

.66198 

.45918 

.67013 

.46788 

.67815 

.47659 

20  20 

9.65382 

0.45063 

9.66212 

0.45932 

9.67027 

0.46802 

9.67828 

0.47674 

24  21 

.65396 

.45077 

.66225 

.45947 

.67040 

.46817 

.67841 

.47688 

2S  22 

.65410 

.45092 

.66239 

.45961 

.67054 

.46831 

.67854 

.47703 

32  23 

.65424 

.45106 

.66253 

.45976 

.67067 

.46846 

.67868 

.47717 

56  24 

.65438 

.45121 

.66266 

.45990 

.67081 

.46860 

.67881 

.47732 

40  25 

9.65452 

0.45135 

9.66280 

0.46005 

9.67094 

0.46875 

9.67894 

0.47746 

44  26 

.65466 

.45150 

.66294 

.46019 

.67108 

.46890 

.67907 

.47761 

45  27 

.65480 

.45164 

.66307 

.46034 

.67121 

.46904 

.67920 

.47775 

52  28 

.65493 

.45179 

.66321 

.46048 

.67134 

.46919 

.67934 

.47790 

55  29 

.65507 

.45193 

.66335 

.46063 

.67148 

.46933 

.67947 

.47805 

s    ' 

5*  38m    84° 

5h  42™    85° 

5^  46™    86° 

5*  50™    87° 

0  30 

9.65521 

0.45208 

9.66348 

0.46077 

9.67161 

0.46948 

9.67960 

0.47819 

4  31 

.65535 

.45222 

.66362 

.46092 

.67175 

.46962 

.67973 

.47834 

8  32 

.65549 

.45237 

.66376 

.46106 

.67188 

.46977 

.67986 

.47848 

72  33 

.65563 

.45251 

.66389 

.46121 

.67202 

.46991 

.68000 

.47863 

76  34 

.65577 

.45266 

.66403 

.46135 

.67215 

.47006 

.68013 

.47877 

20  35 

9.65591 

0.45280 

9.66417 

0.46150 

9.67228 

0.47020 

9.68026 

0.47892 

24  36 

.65605 

.45295 

.66430 

.46164 

.67242 

.47035 

.68039 

.47906 

28  37 

.65619 

.45309 

.66444 

.46179 

.67255 

.47049 

.68052 

.47921 

32  38 

.65632 

.45324 

.66458 

.46193 

.67269 

.47064 

.68066 

.47935 

36  39 

.65646 

.45338 

.66471 

.46208 

.67282 

.47078 

.68079 

.47950 

40  40 

9.65660 

0.45353 

9.66485 

0.46222 

9.67295- 

0.47093 

9.68092 

0.47964 

44  41 

.65674 

.45367 

.66499 

.46237 

.67309 

.47107 

.68105 

.47979 

48  42 

.65688 

.45381 

.66512 

.46251 

.67322 

.47122 

.68118 

.47993 

52  43 

.65702 

.45396 

.66526 

.46266 

.67336 

.47136 

.68131 

.48008 

56  44 

.65716 

.45410 

.66539 

.46280 

.67349 

.47151 

.68144 

.48022 

6'     ' 

5*  39m    84° 

5*  43™    85° 

5h  47m    86° 

5h  S1m     87° 

0  45 

9.65729 

0.45425 

9.66553 

0.46295 

9.67362 

0.47165 

9.68158 

0.48037 

4  46 

.65743 

.45439 

.66567 

.46309 

.67376 

.47180 

.68171 

.48052 

S  47 

.65757 

.45454 

.66580 

.46324 

.67389 

.47194 

.68184 

.48066 

/2  48 

.65771 

.45468 

.66594 

.46338 

.67402 

.47209 

.68197 

.48081 

16  49 

.65785 

.45483 

.66607 

.46353 

.67416 

.47223 

.68210 

.48095 

20  50 

9.65799 

0.45497 

9.66621 

0.46367 

9.67429 

0.47238 

9.68223 

0.48110 

24  51 

.65812 

.45512 

.66635 

.46382 

.67443 

.47252 

.68236 

.48124 

28  52 

.65826 

.45526 

.66648 

.46396 

.67456 

.47267 

.68249 

.48139 

32  53 

.65840 

.45541 

.66662 

.46411 

.67469 

.47282 

.68263 

.48153 

36  54 

.65854 

.45555 

.66675 

.46425 

.67483 

.47296 

.68276 

.48168 

40  55 

9.65868 

0.45570 

9.66689 

0.46440 

9.67496 

0.47311 

9.68289 

0.48182 

44  56 

.65881 

.45584 

.66702 

.46454 

.67509 

.47325 

.68302 

.48197 

48  57 

.65895 

.45599 

.66716 

.46469 

.67522 

.47340 

.68315 

.48211 

52  58 

.65909 

.45613 

.66730 

.46483 

.67536 

.47354 

.68328 

.48226 

56  59 

.65923 

.45628 

.66743 

.46498 

.67549 

.47369 

.68341 

.48241 

60  60 

9.65937 

0.45642 

9.66757 

0.46512 

9.67562 

0.47383 

9.68354 

0.48255 

272 


Table  10.    Haversine  Table 


s   ' 

5h  52m    88° 

5h  56m    89° 

6h  Qm    6^  4™ 

Hav. 

No. 

Hav. 

No. 

S 

Hav. 

Hav. 

0   0 

9.68354 

0.48255 

9.69132 

0.49127 

0 

9.69897 

9.70648 

4   1 

.68367 

.48269 

.69145 

.49142 

4 

.69910 

.70661 

8   2 

.68380 

.48284 

.69158 

.49156 

8 

.69922 

.70673 

12   3 

.68393 

.48299 

.69171 

.49171 

12 

.69935 

.70686 

16   4 

.68407 

.48313 

.69184 

.49186 

16 

.69948 

.70698 

20   5 

9.68420 

0.48328 

9.69197 

0.49200 

20 

9.69960 

9.70710 

24   6 

.68433 

.48342 

.69209 

.49215 

24 

.69973 

.70723 

28   7 

.68446 

.48357 

.69222 

.49229 

28 

.69985 

.70735 

32   8 

.68459 

.48371 

.69235 

.49244 

32 

.69998 

.70748 

36   9 

.68472 

.48386 

.69248 

.49258 

36 

.70011 

.70760 

40  10 

9.68485 

0.48400 

9.69261 

0.49273 

40 

9.70023 

9.70772 

44  11 

.68498 

.48415 

.69274 

.49287 

44 

.70036 

.70785 

45  12 

.68511 

.48429 

.69286 

.49302 

48 

.70048 

.70797 

52  13 

.68524 

.48444 

.69299 

.49316 

52 

.70061 

.70809 

56  14 

.68537 

.48459 

.69312 

.49331 

56 

.70074 

.70822 

s    ' 

5*  53™    88° 

5h  57m    89° 

| 

s 

6h  lm    6h  5m 

0  15 

9.68550 

0.48473 

9.69325 

0.49346 

> 

o3 

0 

9.7008'6 

9.70834 

4  16 

.68563 

.48488 

.69338 

.49360 

A 

4 

.70099 

.70847 

5  17 

.68576 

.48502 

.69350 

.49375 

6 

8 

.70111 

.70859 

12  18 

.68589 

.48517 

.69363 

.49389 

& 

12 

.70124 

.70871 

.76  19 

.68602 

.48531 

.69376 

.49404 

S 

16 

.70136 

.70884 

£0  20 

9.68615 

0.48546 

9.69389 

0.49418 

20 

9.70149 

9.70896 

24  21 

.68628 

.48560 

.69402 

.49433 

3 

24 

.70161 

.70908 

£S  22 

.68641 

.48575 

.69414 

.49447 

A\ 

28 

.70174 

.70921 

32  23 

.68654 

.48589 

.69427 

.49462 

!«• 

32 

.70187 

.70933 

35  24 

.68667 

.48604 

.69440 

.49476 

^§ 

36 

.70199 

.70945 

40  25 

9.68680 

0.48618 

9.69453 

0.49491 

•S  fa 

40 

9.70212 

9.70958 

44  26 

.68693 

.48633 

.69465 

.49506 

3° 

44 

.70224 

.70970 

45  27 

.68706 

.48648 

.69478 

.49520 

5* 

48 

.70237 

.70982 

52  28 

.68719 

.48662 

.69491 

.49535 

°-g 

52 

.70249 

.70995 

55  29 

.68732 

.48677 

.69504 

.49549 

•£  9 
20 

56 

.70262 

.71007 

s    ' 

5^  54m    88° 

5*  58™    89° 

>> 
o>  8 

s 

Qh  %rn     6h  Qm 

0  30 

9.68745 

0.48691 

9.69516 

0.49564 

,8-" 

•+3  rt-l 

0 

9.70274 

9.71019 

4  31 

.68758 

.48706 

.69529 

.49578 

a| 

4 

.70287 

.71032 

8  32 

.68771 

.48720 

.69542 

.49593 

8 

.70299 

.71044 

^2  33 

.68784 

.48735 

.69555 

.49607 

^o  a} 
o  3 

12 

.70312 

.71056 

.76  34 

.68797 

.48749 

.69567 

.49622 

5-2 

16 

.70324 

.71068 

20  35 

9.68810 

0.48764 

9.69580 

0.49636 

is 

20   . 

9.70337 

9.71081 

24  36 

.68823 

.48778 

.69593 

.49651 

£2 

24 

.70349 

.71093 

28  37 

.68836 

.48793 

.69605 

.49665 

"*  03 

28 

.70362 

.71105 

S#  38 

.68849 

.48807 

.69618 

.49680 

R 

32 

.70374 

.71118 

56  39 

.68862 

.48822 

.69631 

.49695 

3 

36 

.70387 

.71130 

40  40 

9.68875 

0.48837 

9.69644 

0.49709 

1 

40 

9.70399 

9.71142 

44  41 

.68887 

.48851 

.69656 

.49724 

• 

44 

.70412 

.71154 

48  42 

.68900 

.48866 

.69669 

.49738 

o 

te 

48 

.70424 

.71167 

52  43 

.68913 

.48880 

.69682 

.49753 

c\ 

52 

.70437 

.71179 

56  44 

.68926 

.48895 

.69694 

.49767 

$ 

~ 

56 

.70449 

.71191 

s    ' 

5h  55m    88° 

5^  59m    89° 

i 

s 

Qh  &*     6h  7m 

0  45 

9.68939 

0.48909 

9.69707 

0.49782 

i 

0 

9.70462 

9.71203 

4  46 

.68952 

.48924 

.69720 

.49796 

i 

4 

.70474 

.71216 

5  47 

.68965 

.48938 

.69732 

.49811 

o 

+7 

8 

.70487 

.71228 

^2  48 

.68978 

.48953 

.69745 

.49825 

f-+ 

12 

.70499 

.71240 

16  49 

.68991 

.48967 

.69758 

.49840 

16 

.70512 

.71252 

#0  50 

9.69004 

0.48982 

9.69770 

0.49855 

20 

9.70524 

9.71265 

24  51 

.69017 

.48997 

.69783 

.49869 

24 

.70537 

.71277 

2<S  52 

.69029 

.49011 

.69796 

.49884 

28 

.70549 

.71289 

32  53 

.69042 

.49026 

.69808 

.49898 

32 

.70561 

.71301 

36  54 

.69055 

.49040 

.69821 

.49913 

36 

.70574 

.71314 

40  55 

9.69068 

0.49055 

9.69834 

0.49927 

40 

9.70586 

9.71326 

44  56 

.69081 

.49069 

.69846 

.49942 

44 

.70599 

.71338 

48  57 

.69094 

.49084 

.69859 

.49956 

48 

.70611 

.71350 

50  58 

.69107 

.49098 

.69872 

.49971 

52 

.70624 

.71362 

56  59 

.69120 

.49113 

.69884 

.49985 

56 

.70636 

.71375 

60  60 

9.69132 

0.49127 

9.69897 

0.50000 

I   60 

9.70648 

9.71387 

Table  10.    Haversine  Table 


273 


g 

Qh  s™   6*  12™ 

6h  IQm    Qh  2Qm 

0*  24m   6^  28m 

6h  32m    Qh  3Qm 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.71387 

9.72112 

9.72825 

9.73526 

9.74215 

9.74891 

9.75556 

9.76209 

4 

.71399 

.72124 

.72837 

.73538 

.74226 

.74902 

.75567 

.76220 

8 

.71411 

.72136 

.72849 

.73549 

.74237 

.74914 

.75578 

.76231 

12 

.71423 

.72148 

.72861 

.73561 

.74249 

.74925 

.75589 

.76241 

16 

.71436 

.72160 

.72873 

.73572 

.74260 

.74936 

.75600 

.76252 

20 

9.71448 

9.72172 

9.72884 

9.73584 

9.74272 

9.74947 

9.75611 

9.76263 

24 

.71460 

.72184 

.72896 

.73596 

.74283 

.74958 

.75622 

.76274 

28 

.71472 

.72196 

.72908 

.73607 

.74294 

.74969 

.75633 

.76285 

82 

.71484 

.72208 

.72920 

.73619 

.74306 

.74981 

.75644 

.76296 

36 

.71496 

.72220 

.72931 

.73630 

.74317 

.74992 

.75655 

.76306 

40 

9.71509 

9.72232 

9.72943 

9.73642 

9.74328 

9.75003 

9.75666 

9.76317 

44 

.71521 

.72244 

.72955 

.73653 

.74340 

.75014 

.75677 

.76328 

48 

.71533 

.72256 

.72967 

.73665 

.74351 

.75025 

.75688 

.76338 

52 

.71545 

.72268 

.72978 

.73676 

.74362 

.75036 

.75698 

.76349 

56 

.71557 

.72280 

.72990 

.73688 

.74374 

.75047 

.75709 

.76360 

s 

Qh  9m    Qh  13m 

Qh  l?m   .  Qh  2lm 

6h  25m   6h  29m 

Qh  ssm   6h  37m 

0 

9.71569 

9.72292 

9.73002 

9.73699 

9.74385 

9.75059 

9.75720 

9.76371 

4 

.71582 

.72304 

.73014 

.73711 

.74396 

.75070 

.75731 

.76381 

8 

.71594 

.72316 

.73025 

.73722 

.74408 

.75081 

.75742 

.76392 

12 

.71606 

.72328 

.73037 

.73734 

.74419 

.75092 

.75753 

.76403 

16 

.71618 

.72340 

.73049 

.73746 

.74430 

.75103 

.75764 

.76414 

20 

9.71630 

9.72352 

9.73060 

9.73757 

9.74442 

9.75114 

9.75775 

9.76424 

24 

.71642 

.72363 

.73072 

.73769 

.74453 

.75125 

.75786 

.76435 

28 

.71654 

.72375 

.73084 

.73780 

.74464 

.75136 

.75797 

.76446 

32 

.71666 

.72387 

.73096 

.73792 

.74475 

.75147 

.75808 

.76456 

36 

.71679 

.72399 

.73107 

.73803 

.74487 

.75159 

.75819 

.76467 

40 

9.71691 

9.72411 

9.73119 

9.73815 

9.74498 

9.75170 

9.75830 

9.76478 

44 

.71703 

.72423 

.73131 

.73826 

.74509 

.75181 

.75840 

.76489 

48 

.71715 

.72435 

.73142 

.73838 

.74521 

.75192 

.75851 

.76499 

52 

.71727 

.72447 

.73154 

.73849 

.74532 

.75203 

.75862 

.76510 

56 

.71739 

.72459 

.73166 

.73860 

.74543 

.75214 

.75873 

.76521 

s 

Qh  iom   6*  14m 

0*  18™   6^  22™ 

0*  26™   6>*  SO"1 

0*  34m   6h  38™ 

0 

9.71751 

9.72471 

9.73177 

9.73872 

9.74554 

9.75225 

9.75884 

9.76531 

4 

.71763 

.72482 

.73189 

.73883 

.74566 

.75236 

.75895 

.76542 

8 

.71775 

.72494 

.73201 

.73895 

.74577 

.75247 

.75906 

.76553 

12 

.71787 

.72506 

.73212 

.73906 

.74588 

.75258 

.75917 

.76563 

16 

.71800 

.72518 

.73224 

.73918 

.74600 

.75269 

.75927 

.76574 

20 

9.71812 

9.72530 

9.73236 

9.73929 

9.74611 

9.75280 

9.75938 

9.76585 

24 

.71824 

.72542 

.73247 

.73941 

.74622 

.75291 

.75949 

.76595 

28 

.71836 

.72554 

.73259 

.73952 

.74633 

.75303 

.75960 

.76606 

32 

.71848 

.72565 

.73271 

.73964 

.74645 

.75314 

.75971 

.76617 

36 

.71860 

.72577 

.73282 

.73975 

.74656 

.75325 

.75982 

.76627 

40 

9.71872 

9.72589 

9.73294 

9.73987 

9.74667 

9.75336 

9.75993 

9.76638 

44 

.71884 

.72601 

.73306 

.73998 

.74678 

.75347 

.76004 

.76649 

48 

.71896 

.72613 

.73317 

.74009 

.74690 

.75358 

.76014 

.76659 

52 

.71908 

.72625 

.73329 

.74021 

.74701 

.75369 

.76025 

.76670 

66 

.71920 

.72637 

.73341 

.74032 

.74712 

.75380 

.76036 

.76681 

8 

0*  llm   6*  15m 

Qh  igm.   Qh  23™ 

6h  27m   6h  31m 

Qh  S5m   0*  39™ 

0 

9.71932 

9.72648 

9.73352 

9.74044 

9.74723 

9.75391 

9.76047 

9.76691 

4 

.71944 

.72660 

.73364 

.74055 

.74734 

.75402 

.76058 

.76702 

8 

.71956 

.72672 

.73375 

.74067 

.74746 

.75413 

.76069 

.76713 

12 

.71968 

.72684 

.73387 

.74078 

.74757 

.75424 

.76079 

.76723 

16 

.71980 

.72696 

.73399 

.74089 

.74768 

.75435 

.76090 

.76734 

20 

9.71992 

9.72708 

9.73410 

9.74101 

9.74779 

9.75446 

9.76101 

9.76745 

24 

.72004 

.72719 

.73422 

.74112 

.74791 

.75457 

.76112 

.76755 

28 

.72016 

.72731 

.73433 

.74124 

.74802 

.75468 

.76123 

.76766 

32 

.72028 

.72743 

.73445 

.74135 

.74813 

.75479 

.76134 

.76777 

36 

.72040 

.72755 

.73457 

.74146 

.74824 

.75490 

.76144 

.76787 

40 

9.72052 

9.72767 

9.73468 

9.74158 

9.74835 

9.75501 

9.76155 

9.76798 

44 

.72064 

.72778 

.73480 

.74169 

.74846 

.75512 

.76166 

.76808 

48 

.72070 

.72790 

.73491 

.74181 

.74858 

.75523 

.76177 

.76819 

52 

.72088 

.72802 

.73503 

.74192 

.74869 

.75534 

.76188 

.76830 

56 

.72100 

.72814 

.73515 

.74203 

.74880 

.75545 

.76198 

.76840 

60 

9.72112 

9.72825 

9.73526 

9.74215 

9.74891 

9.75556 

9.76209 

9.76851 

274 


Table  10.    Haversine  Table 


s 

6h  40m  6h  44m 

Qh  48™  6^  52™ 

Qh  5Qm    7h  Qm 

7h  4™    7h  8™ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.76851 

9.77481 

9.78101 

9.78709 

9.79306 

9.79893 

9.80470 

9.81036 

4 

.76861 

.77492 

.78111 

.78719 

.79316 

.79903 

.80479 

.81045 

8 

.76872 

.77502 

.78121 

.78729 

.79326 

.79913 

.80489 

.81054 

12 

.76883 

.77512 

.78131 

.78739 

.79336 

.79922 

.80498 

.81064 

16 

.76893 

.77523 

.78141 

.78749 

.79346 

.79932 

.80508 

.81073 

20 

9.76904 

9.77533 

9.78152 

9.78759 

9.79356 

9.79942 

9.80517 

9.81082 

24 

.76914 

.77544 

.78162 

.78769 

.79366 

.79951 

.80527 

.81092 

28 

.76925 

.77554 

.78172 

.78779 

.79376 

.79961 

.80536 

.81101 

32 

.76936 

.77564 

.78182 

.78789 

.79385 

.79971 

.80546 

.81110 

36 

.76946 

.77575 

.78192 

.78799 

.79395 

.79980 

.80555 

.81120 

40 

9.76957 

9.77585 

9.78203 

9.78809 

9.79405 

9.79990 

9.80565 

9.88129 

44 

.76967 

.77596 

.78213 

.78819 

.79415 

.80000 

.80574 

.81138 

48 

.76978 

.77606 

.78223 

.78829 

.79425 

.80009 

.80584 

.81148 

52 

.76988 

.77616 

.78233 

.78839 

.79434 

.80019 

.80593 

.81157 

56 

.76999 

.77627 

.78243 

.78849 

.79444 

.80029 

.80603 

.81166 

s 

Qh  ^im   Qh  £5m 

6h  49™  6*  53"1 

Qh  gym    yh  im 

7h  5™    7h  9™ 

0 

9.77009 

9.77637 

9.78254 

9.78859 

9.79454 

9.80038 

9,80612 

9.81176 

4 

.77020 

.77647 

.78264 

.78869 

.79464 

.80048 

.80622 

.81185 

8 

.77031 

.77658 

.78274 

.78879 

.79474 

.80058 

.80631 

.81194 

12 

.77041 

.77668 

.78284 

.78889 

.79484 

.80067 

.80641 

.81204 

16 

.77052 

.77679 

.78294 

.78899 

.79493 

.80077 

.80650 

.81213 

20 

9.77062 

9.77689 

9.78305 

9.78909 

9.79503 

9.30087 

9.80660 

9.81222 

24 

.77073 

.77699 

.78315 

.78919 

.79513 

.80096 

.80669 

.81231 

28 

.77083 

.77710 

.78325 

.78929 

.79523 

.80106 

.80678 

.81241 

32 

.77094 

.77720 

.78335 

.78939 

.79533 

.80116 

.80688 

.81250 

36 

.77104 

.77730 

.78345 

.78949 

.79542 

.80125 

.80697 

.81259 

40 

9.77115 

9.77741 

9.78355 

9.78959 

9.79552 

9.80135 

9.80707 

9.81269 

44 

.77125 

.77751 

.78365 

.78969 

.79562 

.80144 

.80716 

.81278 

48 

.77136 

.77761 

.78376 

.78979 

.79572 

.80154 

.80726 

.81287 

62 

.77146 

.77772 

.78386 

.78989 

.79582 

.80164 

.80735 

.81296 

66 

.77157 

.77782 

.78396 

.78999 

.79591 

.80173 

.80745 

.81306 

s 

6h  42™  6^  46™ 

6h  50™  6>>  54™ 

Qh  QS™   7*  2™ 

7h  6™   7h  10™ 

0 

9.77167 

9.77792 

9.78406 

9.79009 

9.79601 

9.80183 

9.80754 

9.81315 

4 

.77178 

.77803 

.78416 

.79019 

.79611 

.80192 

.80763 

.81324 

8 

.77188 

.77813 

.78426 

.79029 

.79621 

.80202 

.80773 

.81333 

12 

.77199 

.77823 

.78436 

.79039 

.79631 

.80212 

.80782 

.81343 

16  . 

.77209 

.77834 

.78447 

.79049 

.79640 

.80221 

.80792 

.81352 

20 

9.77220 

9.77844 

9.78457 

9.79059 

9.79650 

9.80231 

9.80801 

9.81361 

24 

.77230 

.77854 

.78467 

.79069 

.79660 

.80240 

.80811 

.81370 

28 

.77241 

.77864 

.78477 

.79079 

.79670 

.80250 

.80820 

.81380 

32 

.77251 

.77875 

.78487 

.79089 

.79679 

.80260 

.80829 

.81389 

36 

.77262 

.77885 

.78497 

.79099 

.79689 

.80269 

.80839 

.81398 

40 

9.77272 

9.77895 

9.78507 

9.79108 

9.79699 

9.80279 

9.80848 

9.81407 

44 

.77283 

.77906 

.78517 

.79118 

.79709 

.80288 

.80858 

.81417 

48 

.77293 

.77916 

.78528 

.79128 

.79718 

.80298 

.80867 

.81426 

52 

.77304 

.77926 

.78538 

.79138 

.79728 

.80307 

.80876 

.81435 

56 

.77314 

.77936 

.78548 

.79148 

.79738 

.80317 

.80886 

.81444 

s 

6h  43m  6h  47m 

Qh  Qim   6h  55m 

Qh  59m    7h  S™ 

?h  jrn      7h  Um 

0 

9.77325 

9.77947 

9.78558 

9.79158 

9.79748 

9.80327 

9.80895 

9.81454 

4 

.77335 

.77957 

.78568 

.79168 

.79757 

.80336 

.80905 

.81463 

8 

.77346 

.77967 

.78578 

.79178 

.79767 

.80346 

.80914 

.81472 

12 

.77356 

.77978 

.78588 

.79188 

.79777 

.80355 

.80923 

.81481 

16 

.77366 

.77988 

.78598 

.79198 

.79787 

.80365 

.80933 

.81490 

20 

9.77377 

9.77998 

9.78608 

9.79208 

9.79796 

9.80374 

9.80942 

9.81500 

24 

.77387 

.78008 

.78618 

.79217 

.79806 

.80384 

.80952 

.81509 

28 

.77398 

.78019 

.78628 

.79227 

.79816 

.80393 

.80961 

.81518 

32 

.77408 

.78029 

.78638 

.79237 

.79825 

.80403 

.80970 

.81527 

36 

.77419 

.78039 

.78649 

.79247 

.79835 

.80413 

.80980 

.81536 

40 

9.77429 

9.78049 

9.78659 

9.79257 

9.79845 

9.80422 

9.80989 

9.81546 

44 

.77440 

.78060 

.78669 

.79267 

.79855 

.80432 

.80998 

.81555 

48 

.77450 

.78070 

.78679 

.79277 

.79864 

.80441 

.81008 

.81564 

52 

.77460 

.78080 

.78689 

.79287 

.79874 

.80451 

.81017 

.81573 

56 

.77471 

.78090 

.78699 

.79297 

.79884 

.80460 

.81026 

.81582 

60 

9.77481 

9.78101 

9.78709 

9.79306 

9.79893 

9.80470 

9.81036 

9.81592 

Table  10.    Haversine  Table 


275 


s 

7h  12m  7h  16™ 

7h  20m  7h  24m 

7h  28™  7h  82™ 

7h  36™  7h  40m 

HOT. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.81592 

9.82137 

9.82673 

9.83199 

9.83715 

9.84221 

9.84718 

9.85206 

4 

.81601 

.82146 

.82682 

.83207 

.83723 

.84230 

.84726 

.85214 

8 

.81610 

.82155 

.82691 

.83216 

.83732 

.84238 

.84735 

.85222 

12 

.81619 

.82164 

.82699 

.83225 

.83740 

.84246 

.84743 

.85230 

16 

.81628 

.82173 

.82708 

.83233 

.83749 

.84255 

.84751 

.85238 

20 

9.81637 

9.82182 

9.82717 

9.83242 

9.83757 

9.84263 

9.84759 

9.85246 

24 

.81647 

.82191 

.82726 

.83251 

.83766 

.84271 

.84767 

.85254 

28 

.81656 

.82200 

.82735 

.83259 

.83774 

.84280 

.84776 

.85262 

32 

.81665 

.82209 

.82744 

.83268 

.83783 

.84288 

.84784 

.85270 

36 

.81674 

.82218 

.82752 

.83277 

.83791 

.84296 

.84792 

.85278 

40 

9.81683 

9.82227 

9.82761 

9.83285 

9.83800 

9.84305 

9.84800 

9.85286 

44 

.81692 

.82236 

.82770 

.83294 

.83808 

.84313 

.84808 

.85294 

48 

.81701 

.82245 

.82779 

.83303 

.83817 

.84321 

.84817 

.85302 

62 

.81711 

.82254 

.82788 

.83311 

.83825 

.84330 

.84825 

.85310 

56 

'.81720 

.S2263 

.82796 

.83320 

.83834 

.84338 

.84833 

.85318 

s 

7*  ism  7h  17m 

?h  21m  7*  25m 

7h  29m  7h  33m 

?h  3?m   7h  ^im 

0 

9.81729 

9.82272 

9.82805 

9.83329 

9.83842 

9.84346 

9.84841 

9.85326 

4 

.81738 

.82281 

.82814 

.83337 

.83851 

.84355 

.84849 

.85334 

8 

.81747 

.82290 

.82823 

.83346 

.83859 

.84363 

.84857 

.85342 

12 

.81756 

.82299 

.82832 

.83355 

.83868 

.84371 

.84866 

.85350 

16 

.81765 

.82308 

.82840 

.83363 

.83876 

.84380 

.84874 

.85358 

20 

9.81775 

9.82317 

9.82849 

9.83372 

9.83885 

9.84388 

9.84882 

9.85366 

24 

.81784 

.82326 

.82858 

.83380 

.83893 

.84396 

.84890 

.85374 

28 

.81793 

.82335 

.82867 

.83389 

.83902 

.84405 

.84898 

.85382 

32 

.81802 

.82344 

.82876 

.83398 

.83910 

.84413 

.84906 

.85390 

36 

.81811 

.82353 

.82884 

.83406 

.83919 

.84421 

.84914 

.85398 

40 

9.81820 

9.82362 

9.82893 

9.83415 

9.83927 

9.84430 

9.84923 

9.85406 

44 

.81829 

.82371 

.82902 

.83424 

.83935 

.84438 

.84931 

.85414 

48 

.81838 

.82380 

.82911 

.83432 

.83944 

.84446 

.84939 

.85422 

52 

.81847 

.82388 

.82920 

.83441 

.83952 

.84454 

.84947 

.85430 

56 

.81857 

.82397 

.82928 

.83449 

.83961 

.84463 

.84955 

.85438 

s 

7h  14m  7h  18m 

?h  22m  7h  26™ 

7*  30m  7h  34m 

7*  38m  7*  42m 

0 

9.81866 

9.82406 

9.82937 

9.83458 

9.83969 

9.84471 

9.84963 

9.85446 

4 

.81875 

.82415 

.82946 

.83467 

.83978 

.84479 

.84971 

.85454 

8 

.81884 

.82424 

.82955 

.83475 

.83986 

.84488 

.84979 

.85462 

12 

.81893 

.82433 

.82963 

.83484 

.83995 

.84496 

.84988 

.85470 

16 

.81902 

.82442 

.82972 

.83492 

.84003 

.84504 

.84996 

.85478 

20 

9.81911 

9.82451 

9.82981 

9.83501 

9.84011 

9.84512 

9.85004 

9.85486 

24 

.81920 

.82460 

.82990 

.83510 

.84020 

.84521 

.85012 

.85494 

28 

.81929 

.82469 

.82998 

.83518 

•  .84028 

.84529 

.85020 

.85502 

32 

.81938 

.82478 

.83007 

.83527 

.84037 

.84537 

.85028 

.85510 

36 

.81947 

.82487 

.83016 

.83535 

.84045 

.84545 

.85036 

.85518 

40 

9.81956 

9.82495 

9.83025 

9.83544 

9.84054 

9.84554 

9.85044 

9.85526 

44 

.81965 

.82504 

.83033 

.83552 

.84062 

.84562 

.85052 

.85534 

48 

.81975 

.82513 

.83042 

.83561 

.84070 

.84570 

.85061 

.85542 

52 

.81984 

.82522 

.83051 

.83570 

.84079 

.84578 

.85069 

.85550 

66 

.81993 

.82531 

.83059 

.83578 

.84087 

.84587 

.85077 

.85557 

8 

?h  jsm  7h  19m 

7/»  23nt  7h  27m 

7h  31m  7h  35m 

7h  ggm   ?h  43™ 

0 

9.82002 

9.82540 

9.83068 

9.83587 

9.84096 

(.).S4o«,).J 

9.85085 

9.85565 

4 

.82011 

.82549 

.83077 

.83595 

.84104 

.84603 

.85093 

.85573 

8 

.82020 

.82558 

.83086 

.83604 

.84112 

.84611 

.85101 

.85581 

12 

.82029 

.82567 

.83094 

.83612 

.84121 

.84620 

.85109 

.85589 

16 

.82038 

.82575 

.83103 

.83621 

.84129 

.84628 

.85117 

.85597 

20 

9.82047 

9.82584 

9.83112 

9.83630 

9.84138 

9.84636 

9.85125 

9.85605 

24 

.82056 

.82593 

.83120 

.83638 

.84146 

.84644 

.85133 

.85613 

28 

.82065 

.82602 

.83129 

.83647 

.84154 

.84653 

.85141 

.85621 

32 

.82074 

.82611 

.83138 

.83655 

.84163 

.84661 

.85149 

.85629 

36 

.82083 

.82620 

.83147 

.83664 

.84171 

.84669 

.85158 

.85637 

40 

9.82092 

9.82629 

9.83155 

9.83672 

9.84179 

9.84677 

9.85166 

9.85645 

44 

.82101 

.82638 

.83164 

.83681 

.84188 

.84685 

.85174 

.85653 

48 

.82110 

.82646 

.83173 

.83689 

.84196 

.84694 

.85182 

.85660 

52 

.82119 

.82655 

.83181 

.83698 

.84205 

.84702 

.85190 

.85668 

56 

.82128 

.82664 

.83190 

.83706 

.84213 

.84710 

.85198 

.85676 

60 

9.82137 

9.82673 

9.83199 

9.83715 

9.84221 

9.84718 

9.85206 

9.85684 

276 


Table  10.    Haversine  Table 


s 

7h  ^m   7h  48™ 

7h  ggm   7h  5Qm 

gh  Qm   8*  4m 

8h  gm   8h  12™ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.85684 

9.86153 

9.86613 

9.87064 

9.87506 

9.87939 

9.88364 

9.88780 

4 

.85692 

.86161 

.86621 

.87072 

.87513 

.87947 

.88371 

.88787 

8 

.85700 

.86169 

.86628 

.87079 

.87521 

.87954 

.88378 

.88793 

12 

.85708 

.86176 

.86636 

.87086 

.87528 

.87961 

.88385 

.88800 

16 

.85716 

.86184 

.86643 

.87094 

.87535 

.87968 

.88392 

.88807 

.   20 

9.85724 

9.86192 

9.86651 

9.87101 

9.87543 

9.87975 

9.88399 

9.88814 

24 

.85731 

.86200 

.86659 

.87109 

.87550 

.87982 

.88406 

.88821 

28 

.85739 

.86207 

.86666 

.87116 

.87557 

.87989 

.88413 

.88828 

32 

.85747 

.86215 

.86674 

.87124 

.87564 

.87996 

.88420 

.88835 

36 

.85755 

.86223 

.86681 

.87131 

.87572 

.88004 

.88427 

.88841 

40 

9.85763 

9.86230 

9.86689 

9.87138 

9.87579 

9.88011 

9.88434 

9.88848 

44 

.85771 

.86238 

.86696 

.87146 

.87586 

.88018 

.88441 

.88855 

48 

.85779 

.86246 

.86704 

.87153 

.87593 

.88025 

.88448 

.88862 

52 

.85787 

.86254 

.86712 

.87161 

.87601 

.88032 

.88455 

.88869 

56 

.85794 

.86261 

.86719 

.87168 

.87608 

.88039 

.88.462 

.88876 

s 

7h  45m  7h  49™ 

7h  53™  7h  57m 

gh  im    8h  gm 

gh  Qm   8h  13m 

0 

9.85802 

9.86269 

9.86727 

9.87175 

9.87615 

9.88046 

9.88469 

9.88882 

4 

.85810 

.86277 

.86734 

.87183 

.87623 

.88053 

.88476 

.88889 

8 

.85818 

.86284 

.86742 

.87190 

.87630 

.88061 

.88483 

.88896 

12 

.85826 

.86292 

.86749 

.87198 

.87637 

.88068 

.88490 

.88903 

16 

.85834 

.86300 

.86757 

.87205 

.87644 

.88075 

.88496 

.88910 

20 

9.85841 

9.86307 

9.86764 

9.87212 

9.87652 

9.88082 

9.88503 

9.88916 

24 

.85849 

.86315 

.86772 

.87220 

.87659 

.88089 

.88510 

.88923 

.85857 

.86323 

.86780 

.87227 

.87666 

.88096 

.88517 

.88930 

32 

.85865 

.86331 

.86787 

.87235 

.87673 

.88103 

.88524 

.88937 

36 

.85873 

.86338 

.86795 

.87242 

.87680 

.88110 

.88531 

.88944 

40 

9.85881 

9.86346 

9.86802 

9.87249 

9.87688 

9.88117 

9.88528 

9.88950 

44 

.85888 

.86354 

.86810 

.87257 

.87695 

.88124 

.88545 

.88957 

48 

.85896 

.86361 

.86817 

.87264 

.87702 

.88131 

.88552 

.88964 

52 

.85904 

.86369 

.86825 

.87271 

.87709 

.88139 

.88559 

.88971 

56 

.85912 

.86377 

.86832 

.87279 

.87717 

.88146 

.88566 

.88978 

s 

7h  46m  7h  50m 

7h  54m  7h  58m 

8h  2™   8h  6™ 

8h  iom  8^  14m 

0 

9.85920 

9.86384 

9.86840 

9.87286 

9.87724 

9.88153 

9.88573 

9.88984 

4 

.85928 

.86392 

.86847 

.87294 

.87731 

.88160 

.88580 

.88991 

8 

.85935 

.86400 

.86855 

.87301 

.87738 

.88167 

.88587 

.88998 

12 

.85943 

.86407 

.86862 

.87308 

.87745 

.88174 

.88594 

.89005 

16 

.85951 

.86415 

.86870 

.87316 

.87753 

.88181 

.88600 

.89012 

20 

9.85959 

9.86423 

9.86877 

9.87323 

9.87760 

9.88188 

9.88607 

9.89018 

24 

.85967 

.86430 

.86885 

.87330 

.87767 

.88195 

.88614 

.89025 

28 

.85974 

.86438 

.86892 

.87338 

.87774 

.88202 

.88621 

.89032 

32 

.85982 

.86446 

.86900 

.87345 

.87782 

.88209 

.88628 

.89039 

36 

.85990 

.86453 

.86907 

.87352 

.87789 

.88216 

.88635 

.89045 

40 

9.85998 

9.86461 

9.86915 

9.87360 

9.87796 

9.88223 

9.88642 

9.89052 

44 

.86006 

.86468 

.86922 

.87367 

.87803 

.88230 

.88649 

.89059 

48 

.86013 

.86476 

.86930 

.87374 

.87810 

.88237 

.88656 

.89066 

52 

.86021 

.86484 

.86937 

.87382 

.87818 

.88244 

.88663 

.89072 

56 

.86029 

.86491 

.86945 

.87389 

.87825 

.88252 

.88670 

.89079 

s 

7h  47m  7h  51m 

7h  55m  7h  59m 

gh  grn    8h  7m 

8h  llm  8h  15™ 

0 

9.86037 

9.86499 

9.86952 

9.87396 

9.87832 

9.88259 

9.88677 

9.89086 

4 

.86045 

.86507 

.86960 

.87404 

.87839 

.88266 

.88683 

.89093 

8 

.86052 

.86514 

.86967 

.87411 

.87846 

.88273 

.88690 

.89099 

12 

.86060 

.86522 

.86975 

.87418 

.87853 

.88280 

.88697 

.89106 

16 

.86068 

.86529 

.86982 

.87426 

.87861 

.88287 

.88704 

.89113 

20 

9.86076 

9.86537 

9.86990 

9.87433 

9.87868 

9.88294 

9.88711 

9.89120 

24 

.86083 

.86545 

.86997 

.87440 

.87875 

.88301 

.88718 

.89126 

28 

.86091 

.86552 

.87004 

.87448 

.87882 

.88308 

.88725 

.89133 

32 

.86099 

.86560 

.87012 

.87455 

.87889 

.88315 

.88732 

.89140 

36 

.86107 

.86568 

.87019 

.87462 

.87896 

.88322 

.88739 

.89147 

40 

9.86114 

9.86575 

9.87027 

9.87470 

9.87904 

9.88329 

9.88745 

9.89153 

44 

.86122 

.86583 

.87034 

.87477 

.87911 

.88336 

.88752 

.89160 

48 

.86130 

.86590 

.87042 

.87484 

.87918 

.88343 

.88759 

.89167 

52 

.86138 

.86598 

.87049 

.87492 

.87925 

.88350 

.88766 

.89174 

66 

.86145 

.86606 

.87057 

.87499 

.87932 

.88357 

.88773 

.89180 

60 

9.86153 

9.86613 

9.87064 

9.87506 

9.87939 

9.88364 

9.88780 

9.89187 

Table  10.    Haversine  Table 


277 


g 

8h  i$m  8*  20" 

8h  24m  8*  28™ 

8h  32m   8h  36m 

8h  40™  8^  44m 

Hay. 

Hav. 

Hav. 

Hay. 

Hay. 

Hav. 

Hav. 

Hav. 

0 

KS91S7 

9.89586 

9.89976 

9.90358 

9.90732 

9.91098 

9.91455 

9.91805 

4 

.89194 

.89592 

.89983 

.90365 

.90738 

.91104 

.91461 

.91810 

8 

.89200 

.89599 

.89989 

.90371 

.90744 

.91110 

.91467 

.91816 

12 

.89207 

.89606 

.89995 

.90377 

.90751 

.91116 

.91473 

.91822 

16 

.89214 

.89612 

.90002 

.90383 

.90757 

.91122 

.91479 

.91828 

20 

9.89221 

9.89619 

9.90008 

9.90390 

9.90763 

9.91128 

9.91485 

9.91833 

24 

.89227 

.89625 

.90015 

.90396 

.90769 

.91134 

.91490 

.91839 

28 

.89234 

.89632 

.90021 

.90402 

.90775 

.91140 

.91496 

.91845 

32 

.89241 

.89638 

.90028 

.90409 

.90781 

.91146 

.91502 

.91851 

36 

.89247 

.89645 

.90034 

.90415 

.90787 

.91152 

.91508 

.91856 

40 

9.89254 

9.89651 

9.90040 

9.90421 

9.90794 

9.91158 

9.91514 

9.91862 

44 

.89261 

.89658 

.90047 

.90428 

.90800 

.91164 

.91520 

.91868 

48 

.89267 

.89665 

.90053 

.90434 

.90806 

.91170 

.91526 

.91874 

52 

.89274 

.89671 

.90060 

.90440 

.90812 

.91176 

.91532 

.91879 

56 

.89281 

.89678 

.90066 

.90446 

.90818 

.91182 

.91537 

.91885 

s 

8h  1?™   gh  21m 

8h  25m  8h  29m 

8h  Sgm   8h  8Jm 

8h  ^im   8h  ^5m 

0 

9.89287 

9.896S4 

9.90072 

9.90452 

9.90824 

9.91188 

9.91543 

9.91891 

4 

.89294 

.89691 

.90079 

.90459 

.90830 

.91194 

.91549 

.91896 

8 

.89301 

.89697 

.90085 

.90465 

.90836 

.91200 

.91555 

.91902 

12 

.89308 

.89704 

.90092 

.90471 

.90843 

.91206 

.91561 

.91908 

16 

.89314 

.89710 

.90098 

.90478 

.90849 

.91212 

.91567 

.91914 

20 

9.89321 

9.89717 

9.90104 

9.90484 

9.90855 

9.91218 

9.91573 

9.91919 

24 

.89328 

.89723 

.90111 

.90490 

.90861 

.91224 

.91578 

.91925 

28 

.89334 

.89730 

.90117 

.90496 

.90867 

.91230 

.91584 

.91931 

32 

.89341 

.89736 

.90124 

.90503 

.90873 

.91236 

.91590 

.91936 

36 

.89348 

.89743 

.90130 

.90509 

.90879 

.91242 

.91596 

.91942 

40 

9.89354 

9.89749 

9.90136 

9.90515 

9.90885 

9.91248 

9.91602 

9.91948 

44 

.89361 

.89756 

.90143 

.90521 

.90892 

.91254 

.91608 

.91954 

48 

.89368 

.89763 

.90149 

.90527 

.90898 

.91260 

.91613 

.91959 

62 

.89374 

.89769 

.90156 

.90534 

.90904 

.91265 

.91619 

.91965 

56 

.89381 

.89776 

.90162 

.90540 

.90910 

.91271 

.91625 

.91971 

s 

8h  i8m  8h  22™ 

gh  26™  8*  30m 

8h  SJm   8h  3S™ 

8h  j^2m  8^  46™ 

0 

9.89387 

9.89782 

9.90168 

9.90546 

9.90916 

9.91277 

9.91631 

9.91976 

4 

.89394 

.89789 

.90175 

.90552 

.90922 

.91283 

.91637 

.91982 

8 

.89400 

.89795 

.90181 

.90559 

.90928 

.91289 

.91643 

.91988 

12 

.89407 

.89802 

.90187 

.90565 

.90934 

.91295 

.91648 

.91993 

16 

.89414 

.89808 

.90194 

.90571 

.90940 

.91301 

.91654 

.91999 

20 

9.89421 

9.89815 

9.90200 

9.90577 

9.90946 

9.91307 

9.91660 

9.92005 

24 

.89427 

.89821 

.90206 

.90584 

.90952 

.91313 

.91666 

.92010 

28 

.89434 

.89828 

.90213 

.90590 

.90958 

.91319 

.91672 

.92016 

32 

.89441 

.89834 

.90219 

.90596 

.90965 

.91325 

.91677 

.92022 

36 

.89447 

.89840 

.90225 

.90602 

.90971 

.91331 

.91683 

.92027 

40 

9.89454 

9.89847 

9.90232 

9.90608 

9.90977 

9.91337 

9.91689 

9.92033 

44 

.89460 

.89853 

.90238 

.90615 

.90983 

.91343 

.91695 

.92039 

48 

.89467 

.89860 

.90244 

.90621 

.90989 

.91349 

.91701 

.92044 

62 

.89474 

.89866 

.90251 

.90627 

.90995 

.91355 

.91706 

.92050 

56 

.89480 

.89873 

.90257 

.90633 

.91001 

.91361 

.91712 

.92056 

8 

gh  ICfm   gh  23m 

8h  27m  8>>  31m 

8h  S5m  8*  39m 

8*  43™  8*  47m 

0 

9.89487 

9.89879 

9.90264 

9.90639 

9.91007 

9.91367 

9.91718 

9.92061 

4 

.89493 

.89886 

.90270 

.90646 

.91013 

.91372 

.91724 

.92067 

8 

.89500 

.89<S92 

.90276 

.90652 

.91019 

.91378 

.91730 

.92073 

12 

.89507 

.89899 

.90282 

.90658 

.91025 

.91384 

.91735 

.92078 

16 

.89513 

.89905 

.90289 

.90664 

.91031 

.91390 

.91741 

.92084 

20 

9.89520 

9.89912 

9.90295 

9.90670 

9.91037 

9.91396 

9.91747 

9.92090 

24 

.89527 

.89918 

.90301 

.90676 

.91043 

.91402 

.91753 

.92095 

28 

.89533 

.89925 

.90308 

.90683 

.91049 

.91408 

.91758 

.92101 

32 

.89540 

.89931 

.90314 

.90689 

.91055 

.91414 

.91764 

.92107 

36 

.89546 

.89938 

.90320 

.90695 

.91061 

.91420 

.91770 

.92112 

40 

9.89553 

9.89944 

9.90327 

9.90701 

9.01067 

9.91426 

9.91776 

9.92118 

44 

.89559 

.89950 

.90333 

.90707 

.91074 

.91432 

.91782 

.92124 

48 

.89566 

.89957 

.90339 

.90714 

.91080 

.91437 

.91787 

.92129 

62 

.89573 

.89963 

.90346 

.90720 

.91086 

.91443 

.91793 

.92135 

56 

.89579 

.89970 

.90352 

.90726 

.91092 

.91449 

.91799 

.92140 

60 

9.89586 

9.89976 

9.90358 

9.90732 

9.19098 

9.91455 

9.91805 

9.92146 

278 


Table  10.    Haversine  Table 


s 

8h  48™  8h  52™ 

8h  66m   9h  Om 

9h  yn    Qh  gm 

Qh  Igm   Qh  16m 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.92146 

9.92480 

9.92805 

9.93123 

9.93433 

9.93736 

9.94030 

9.94318 

4 

.92152 

.92485 

.92811 

.93128 

.93438 

.93741 

.94035 

.94322 

8 

.92157 

.92491 

.92816 

.93134 

.93443 

.93746 

.94040 

.94327 

12 

.92163 

.92496 

.92821 

.93139 

.93448 

.93751 

.94045 

.94332 

16 

.92169 

.92502 

.92827 

.93144 

.93454 

.93755 

.94050 

.94336 

20 

9.92174 

9.92507 

9.92832 

9.93149 

9.93459 

9.93760 

9.94055 

9.94341 

24 

.92180 

.92512 

.92837 

.93154 

.93464 

.93765 

.94059 

.94346 

28 

.92185 

.92518 

.92843 

.93160 

.93469 

.93770 

.94064 

.94351 

32 

.92191 

.92523 

.92848 

.93165 

.93474 

.93775 

.94069 

.94355 

36 

.92197 

.92529 

.92853 

.93170 

.93479 

.93780 

.94074 

.94360 

40 

9.92202 

9.92534 

9.92859 

9.93175 

9.93484 

9.93785 

9.94079 

9.94365 

44 

.92208 

.92540 

.92864 

.93181 

.93489 

.93790 

.94084 

.94369 

48 

.92213 

.92545 

.92869 

.93186 

.93494 

.93795 

.94088 

.94374 

52 

.92219 

.92551 

.92875 

.93191 

.93499 

.93800 

.94093 

.94379 

56 

.92225 

.92556 

.92880 

.93196 

.93504 

.93805 

.94098 

.94383 

s 

8h  Jfim   gh  53m 

8h  57™   9h  jm 

Qh  5m    Qh  Qm 

Qh  ISm   Qh  17m 

0 

9.92230 

9.92562 

9.92885 

9.93201 

9.93509 

9.93810 

9.94103 

9.94388 

4 

.92236 

.92567 

.92891 

.93207 

.93515 

.93815 

.94108 

.94393 

8 

.92241 

.92573 

.92896 

.93212 

.93520 

.93820 

.94112 

.94398 

12 

.92247 

.92578 

.92901 

.93217 

.93525 

.93825 

.94117 

.94402 

16 

.92253 

.92584 

.92907 

.93222 

.93530 

.93830 

.94122 

.94407 

20 

9.92258 

9.92589 

9.92912 

9.93227 

9.93535 

9.93835 

9.94127 

9.94412 

24 

.92264 

.92594 

.92917 

.93232 

.93540 

.93840 

.94132 

.94416 

28 

.92269 

.92600 

.92923 

.93238 

.93545 

.93845 

.94137 

.94421 

32 

.92275 

.92605 

.92928 

.93243 

.93550 

.93849 

.94141 

.94426 

36 

.92280 

.92611 

.92933 

.93248 

.93555 

.93854 

.94146 

.94430 

40 

9.92286 

9.92616 

9.92939 

9.93253 

9.93560 

9.93859 

9.94151 

9.94435 

44 

.92292 

.92622 

.92944 

.93258 

.93565 

.93864 

.94156 

.94440 

48 

.92297 

.92627 

.92949 

.93264 

.93570 

.93869 

.94161 

.94444 

62 

.92303 

.92633 

.92955 

.93269 

.93575 

.93874 

.94165 

.94449 

66 

.92308 

.92638 

.92960 

.93274 

.93580 

.93879 

.94170 

.94454 

s 

8h  gom  8^  54m 

8h  58™   9h  2™ 

Qh  em   Qh  1Qm 

(jh  Ufm   Qh  18m 

0 

9.92314 

9.92643 

9.92965 

9.93279 

9.93585 

9.93884 

9.94175 

9.94458 

4 

.92319 

.92649 

.92970 

.93284 

.93590 

.93889 

.94180 

.94463 

8 

.92325 

.92654 

.92975 

.93289 

.93595 

.93894 

.94184 

.94468 

12 

.92330 

.92660 

.92981 

.93295 

.93600 

.93899 

.94189 

.94472 

16 

.92336 

.92665 

.92986 

.93300 

.93605 

.93904 

.94194 

.94477 

20 

9.92342 

9.92670 

9.92992 

9.93305 

9.93611 

9.93908 

9.94199 

9.94482 

24 

.92347 

.92676 

.92997 

.93310 

.93616 

.93913 

.94204 

.94486 

28 

.92353 

.92681 

.93002 

.93315 

.93621 

.93918 

.94208 

.94491 

32 

.92358 

.92687 

.93007 

.93320 

.93626 

.93923 

.94213 

.94496 

36 

.92364 

.92692 

.93013 

.93326 

.93631 

.93928 

.94218 

.94500 

40 

9.92369 

9.92698 

9.93018 

9.93331 

9.93636 

9.93933 

9.94223 

9.94505 

44 

.92375 

.92703 

.93023 

.93336 

.93641 

.93938 

.94227 

.94509 

48 

.92380 

.92708 

.93029 

.93341 

.93646 

.93943 

.94232 

.94514 

62 

.92386 

.92714 

.93034 

.93346 

.93651 

.93948 

.94237 

.94519 

66 

.92391 

.92719 

.93039 

.93351 

.93656 

.93952 

.94242 

.94523 

s 

8h  61m  8h  65m 

8h  QQm    Qh  gm 

Qh  jm   gh  nm 

Qh  IQm.   Qh  IQm 

0 

9.92397 

9.92725 

9.93044 

9.93356 

9.93661 

9.93957 

9.94246 

9.94528 

4 

.92402 

.92730 

.93050 

.93362 

.93666 

.93962 

.94251 

.94533 

8 

.92408 

.92735 

.93055 

.93367 

.93671 

.93967 

.94256 

.94537 

12 

.92413 

.92741 

.93060 

.93372 

.93676 

.93972 

.94261 

.94542 

16 

.92419 

.92746 

.93065 

.93377 

.93681 

.93977 

.94265 

.94546 

20 

9.92425 

9.92751 

9.93071 

9.93382 

9.93686 

9.93982 

9.94270 

9.94551 

24 

.92430 

.92757 

.93076 

.93387 

.93691 

.93987 

.94275 

.94556 

28 

.92436 

.92762 

.93081 

.93392 

.93696 

.93991 

.94280 

.94560 

32 

.92441 

.92768 

.93086 

.93397 

.93701 

.93996 

.94284 

.94565 

36 

.92447 

.92773 

.93092 

.93403 

.93706 

.94001 

.94289 

.94570 

40 

9.92452 

9.92778 

9.93097 

9.93408 

9.93711 

9.94006 

9.94294 

9.94574 

44 

.92458 

.92784 

.93102 

.93413 

.93716 

.94011' 

.94299 

.94579 

48 

.92463 

.92789 

.93107 

.93418 

.93721 

.94016 

.94303 

.94583 

62 

.92469 

.92794 

.93113 

.93423 

.93726 

.94021 

.94308 

.94588 

66 

.92474 

.92800 

.93118 

.93428 

.93731 

.94026 

.94313 

.94593 

60 

9.92480 

9.92805 

9.93123 

9.93433 

9.93736 

9.94030 

9.94318 

9.94597 

Table  10.    Haversine  Table 


279 


s 

Qh  20m  9*  24m 

Qh  28m   Qh  3%n 

9*  36™  9*  40m 

9*  44m  9*  48™ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.94597 

9.94869 

9.95134 

9.95391 

9.95641 

9.95884 

9.96119 

9.96347 

4 

.94602 

.94874 

.95138 

.95396 

.95645 

.95888 

.96123 

.96351 

8 

.94606 

.94878 

.95143 

.95400 

.95649 

.95892 

.96127 

.96355 

12 

.94611 

.94883 

.95147 

.95404 

.95654 

.95896 

.96131 

.96359 

16 

.94616 

.94887 

.95151 

.95408 

.95658 

.95900 

.96135 

.96362 

20 

9.94620 

9.94892 

9.95156 

9.95412 

9.95662 

9.95904 

9.96139 

9.96366 

24 

.94625 

.94896 

.95160 

.95417 

.95666 

.95908 

.96142 

.96370 

28 

.94629 

.94901 

.95164 

.95421 

.95670 

.95912 

.96146 

.96374 

82 

.94634 

.94905 

.95169 

.95425 

.95674 

.95916 

.96150 

.96377 

36 

.94638 

.94909 

.95173 

.95429 

.95678 

.95920 

.96154 

.96381 

40 

9.94643 

9.94914 

9.95177 

9.95433 

9.95682 

9.95924 

9.96158 

9.96385 

44 

.94648 

.94918 

.95182 

.95438 

.95686 

.95928 

.96162 

.96388 

48 

.94652 

.94923 

.95186 

.95442 

.95690 

.95932 

.96165 

.96392 

52 

.94657 

.94927 

.95190 

.95446 

.95694 

.95936 

.96169 

.96396 

66 

.94661 

.94932 

.95195 

.95450 

.95699 

.95939 

.96173 

.96400 

s 

Qh  21m  9*  25m 

Qh  gCfm   Qh  3gm 

Qh  37m   Qh  4im 

9*  45m  9h  49m 

0 

9.94666 

9.94936 

9.95199 

9.95454 

9.95703 

9.95943 

9.96177 

9.96403 

4 

.94670 

.94941 

.95203 

.95459 

.95707 

.95947 

.96181 

.96407 

8 

.94675 

.94945 

.95208 

.95463 

.95711 

.95951 

.96185 

.96411 

12 

.94680 

.94950 

.95212 

.95467 

.95715 

.95955 

.96188 

.96412 

16 

.94684 

.94954 

.95216 

.95471 

.95719 

.95959 

.96192 

.96418 

20 

9.94689 

9.94958 

9.95221 

9.95475 

9.95723 

9.95963 

9.96196 

9.96422 

24 

.94693 

.94963 

.95225 

.95480 

.95727 

.95967 

.96200 

.96426 

28 

.94698 

.94967 

.95229 

.95484 

.95731 

.95971 

.96204 

.96429 

32 

.94702 

.94972 

.95234 

.95488 

.95735 

.95975 

.96208 

.96433 

36 

.94707 

.94976 

.95238 

.95492 

.95739 

.95979 

.96211 

.96437 

40 

9.94711 

9.94981 

9.95242 

9.95496 

9.95743 

9.95983 

9.96215 

9.96440 

44 

.94716 

.94985 

.95246 

.95501 

.95747 

.95987 

.96219 

.96444 

48 

.94721 

.94989 

.95251 

.95505 

.95751 

.95991 

.96223 

.96448 

52 

.94725 

.94994 

.95255 

.95509 

.95755 

.95995 

.96227 

.96451 

56 

.94730 

.94998 

.95259 

.95513 

.95759 

.95999 

.96230 

.96455 

s 

9*  22™  9*  26™ 

9h  QQm   Qh  S4n 

9h  ss™  9*  42™ 

Qh  J^Qrn   Qh  QQm 

0 

9.94734 

9.95003 

9.95264 

9.95517 

9.95763 

9.96002 

9.96234 

9.96459 

4 

.94739 

.95007 

.95268 

.95521 

.95768 

.96006 

.96238 

.96462 

8 

.94743 

.95011 

.95272 

.95526 

.95772 

.96010 

.96242 

.96466 

12 

.94748 

.95016 

.95276 

.95530 

.95776 

.96014 

.96246 

.96470 

16 

.94752 

.95020 

.95281 

.95534 

.95780 

.96018 

.96249 

.96473 

20 

9.94757 

9.95025 

9.95285 

9.95538 

9.95784 

9.96022 

9.96253 

9.96477 

24 

.94761 

.95029 

.95289 

.95542 

.95788 

.96026 

.96257 

.96481 

28 

.94766 

.95033 

.95294 

.95546 

.95792 

.96030 

.96261 

.96484 

32 

.94770 

.95038 

.95298 

.95550 

.95796 

.96034 

.96265 

.96488 

36 

.94774 

.95042 

.95302 

.95555 

.95800 

.96038 

.96268 

.96492 

40 

9.94779 

9.95047 

9.95306 

9.95559 

9.95804 

9.96042 

9.96272 

9.96495 

44 

.94784 

.95051 

.95311 

.95563 

.95808 

.96046 

.96276 

.96499 

48 

.94788 

.95055 

.95315 

.95567 

.95812 

.96049 

.96280 

.96503 

52 

.94793 

.95060 

.95319 

.95571 

.95816 

.96053 

.96283 

.96506 

56 

.94797 

.95064 

.95323 

.95575 

.95820 

.96057 

.96287 

.96510 

s 

Qh  ggm   Qh  gym 

Qh  3l>n   Qh  36m 

Qh  SQm   Qh  43m 

9*  47m  9h  51™ 

0 

9.94802 

9.95069 

9.95328 

9.95579 

9.95824 

9.96061 

9.96291 

9.96514 

4 

.94806 

.95073 

.95332 

.95584 

.95828 

.96065 

.96295 

.96517 

8 

.94811 

.95077 

.95336 

.95588 

.95832 

.96069 

.96299 

.96521 

12 

.94815 

.95082 

.95340 

.95592 

.95836 

.96073 

.96302 

.96525 

16 

.94820 

.95086 

.95345 

.95596 

.95840 

.96077 

.96306 

.96528 

20 

9.94824 

9.95090 

9.95349 

9.95600 

9.95844 

9.96081 

9.96310 

9.96532 

24 

.94829 

.95095 

.95353 

.95604 

.95848 

.96084 

.96314 

.96536 

28 

.94833 

.95099 

.95357 

.95608 

.95852 

.96088 

.96317 

.96539 

32 

.94838 

.95104 

.95362 

.95613 

.95856 

.96092 

.96321 

.96543 

36 

.94842 

.95108 

.95366 

.95617 

.95860 

.96096 

.96325 

.96547 

40 

9.94847 

9.95112 

9.95370 

9.95621 

9.95864 

9.96100 

9.96329 

9.96550 

44 

.94851 

.95117 

.95374 

.95625 

.95868 

.96104 

.96332 

.96554 

48 

.94856 

.95121 

.95379 

.95629 

.95872 

.96108 

.96336 

.96557 

52 

.94860 

.95125 

.95383 

.95633 

.95876 

.96112 

.96340 

.96561 

56 

.94865 

.95130 

.95387 

.95637 

.95880 

.96115 

.96344 

.96565 

60 

9.94869 

9.95134 

9.95391 

9.95641 

9.95884 

9.96119 

9.96347 

9.96568 

280 


Table  10.    Haversine  Table 


s 

Qh  5%™   gh  ,5gm 

10*  Om  10*  4m 

10h  8m  10h  12m 

IQh  16™  10*  20m 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.96568 

9.96782 

9.96989 

9.97188 

9.97381 

9.97566 

9.97745 

9.97916 

4 

.96572 

.96786 

.96992 

.97192 

.97384 

.97569 

.97748 

.97919 

8 

.96576 

.96789 

.96996 

.97195 

.97387 

.97572 

.97751 

.97922 

12 

.96579 

.96793 

.96999 

.97198 

.97390 

.97575 

.97754 

.97925 

16 

.96583 

.96796 

.97002 

.97201 

.97393 

.97578 

.97756 

.97927 

20 

9.96586 

9.96800 

9.97006 

9.97205 

9.97397 

9.97581 

9.97759 

9.97930 

24 

.96590 

.96803 

.97009 

.97208 

.97400 

.97584 

.97762 

.97933 

28 

.96594 

.96807 

.97012 

.97211 

.97403 

.97587 

.97765 

.97936 

32 

.96597 

.96810 

.97016 

.97214 

.97406 

.97591 

.97768 

.97939 

36 

.96601 

.96814 

.97019 

.97218 

.97409 

.97594 

.97771 

.97941 

40 

9.96604 

9.96817 

9.97022 

9.97221 

9.97412 

9.97597 

9.97774 

9.97944 

44 

.96608 

.96821 

.97026 

.97224 

.97415 

.97600 

.97777 

.97947 

48 

.96612 

.96824 

.97029 

.97227 

.97418 

.97603 

.97780 

.97950 

52 

.96615 

.96827 

.97033 

.97231 

.97422 

.97606 

.97783 

.97953 

56 

.96619 

.96831 

.97036 

.97234 

.97425 

.97609 

.97785 

.97955 

s 

Qh  ggm   Qh  Q^m 

10h  lm  10h  5m 

10*  9m  10*  13m 

10*  17m  10*  21m 

0 

9.96622 

9.96834 

9.97039 

9.97237 

9.97428 

9.97612 

9.97788 

9.97958 

4 

.96626 

.96837 

.97043 

.97240 

.97431 

.97615 

.97791 

.97961 

8 

.96630 

.96841 

.97046 

.97244 

.97434 

.97618 

.97794 

.97964 

12 

.96633 

.96845 

.97049 

.97247 

.97437 

.97621 

.97797 

.97966 

16 

.96637 

.96848 

.97052 

.97250 

.97440 

.97624 

.97800 

.97969 

20 

9.96640 

9.96852 

9.97056 

9.97253 

9.97443 

9.97627 

9.97803 

9.97972 

24 

.96644 

.96855 

.97059 

.97257 

.97447 

.97630 

.97806 

.97975 

28 

.96648 

.96859 

.97063 

.97260 

.97450 

.97633 

.97808 

.97977 

32 

.96651 

.96862 

.97066 

.97263 

.97453 

.97636 

.97811 

.97980 

36 

.96655 

.96866 

.97069 

.97266 

.97456 

.97639 

.97814 

.97983 

40 

9.96658 

9.96869 

9.97073 

9.97269 

9.97459 

9.97642 

9.97817 

9.97986 

44 

.96662 

.96873 

.97076 

.97273 

.97462 

.97645 

.97820 

.97988 

48 

.96665 

.96876 

.97079 

.97276 

.97465 

.97647 

.97823 

.97991 

62 

.96669 

.96879 

.97083 

.97279 

.97468 

.97650 

.97826 

.97994 

56 

.96673 

.96883 

.97086 

.97282 

.97471 

.97653 

.97829 

.97997 

s 

Qh  Qjm   Qh  Q8m 

10*  2™  10*  6™ 

10*  10m  10h  14m 

10*  18m  10*  22m 

0 

9.96676 

9.96886 

9.97089 

9.97285 

9.97474 

9.97656 

9.97831 

9.97999 

4 

.96680 

.96890 

.97093 

.97289 

.94478 

.97659 

.97834 

.98002 

8 

.96683 

.96894 

.97096 

.96292 

.97481 

.97662 

.97837 

.98005 

12 

.96687 

.96897 

.97099 

.97295 

.97484 

.97665 

.97840 

.98008 

16 

.96690 

.96900 

.97103 

.97298 

.97487 

.97668 

.97843 

.98010 

20 

9.96694 

9.96904 

9.97106 

9.97301 

9.97490 

9.97671 

9.97846 

9.98013 

24 

.96697 

.96907 

.97109 

.97305 

.97493 

.97674 

.97849 

.98016 

28 

.96701 

.96910 

.97113 

.97308 

.97496 

.97677 

.97851 

.98019 

32 

.96705 

.96914 

.97116 

.97311 

.97499 

.97680 

.97854 

.98021 

36 

.96708 

.96917 

.97119 

.97314 

.97502 

.97683 

.97857 

.98024 

40 

9.96712 

9.96921 

9.97123 

9.97317 

9.97505 

9.97686 

9.97860 

9.98027 

44 

.96715 

.96924 

.97126 

.97321 

.97508 

.97689 

.97863 

.98030 

48 

.96719 

.96928 

.97129 

.97324 

.97511 

.97692 

.97866 

.98032 

52 

.96722 

.96931 

.97132 

.97327 

.97514 

.97695 

.97868 

.98035 

56 

.96726 

.96934 

.97136 

.97330 

.97518 

.97698 

.97871 

.98038 

s 

gh  55m   Qh  59m 

10h  3m  10*  7m 

10h  llm  10h  15m 

10*  19m  10*  23m 

0 

9.96729 

9.96938 

9.97139 

9.97333 

9.97521 

9.97701 

9.97874 

9.98040 

4 

.96733 

.96941 

.97142 

.97337 

.97524 

.97704 

.97877 

.98043 

8 

.96736 

.96945 

.97146 

.97340 

.97527 

.97707 

.97880 

.98046 

12 

.96740 

.96948 

.97149 

.97343 

.97530 

.97710 

.97883 

.98049 

16 

.96743 

.96951 

.97152 

.97346 

.97533 

.97713 

.97885 

.98051 

20 

9.96747 

9.96955 

9.97156 

9.97349 

9.97536 

9.97716 

9.97888 

9.98054 

24 

.96750 

.96958 

.97159 

.97352 

.97539 

.97718 

.97891 

.98057 

28 

.96754 

.96962 

.97162 

.97356 

.97542 

.97721 

.97894 

.98059 

32 

.96758 

.96965 

.97165 

.97359 

.97545 

.97724 

.97897 

.98062 

36 

.98761 

.96968 

.97169 

.97362 

.97548 

.97727 

.97899 

.98065 

40 

9.96765 

9.96972 

9.97172 

9.97365 

9.97551 

9.97730 

9.97902 

9.98067 

44 

.96768 

.96975 

.97175 

.97368 

.97554 

.97733 

.97905 

.98070 

48 

.96772 

.96979 

.97179 

.97371 

.97557 

.97736 

.97908 

.98073 

52 

.96775 

.96982 

.97182 

.97375 

.97560 

.97739 

.97911 

.98076 

56 

.96779 

.96985 

.97185 

.97378 

.97563 

.97742 

.97914 

.98078 

60 

9.96782 

9.96989 

9.97188 

9.97381 

9.97566 

9.97745 

9.97916 

9.98081 

Table  10.    Haversiiie  Table 


281 


$ 

10*  24m  10*  28m 

10*32™  10*36m 

10*  40m  10*  44m 

10*  48m  10*  52™ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.98081 

9.98239 

9.98389 

9.98533 

9.98670 

9.98801 

9.98924 

9.99041 

4 

.98084 

.98241 

.98392 

.98536 

.98673 

.98803 

.98926 

.99043 

8 

.98086 

.98244 

.98394 

.98538 

.98675 

.98805 

.98928 

.99044 

12 

.98089 

.98246 

.98397 

.98540 

.98677 

.98807 

.98930 

.99046 

16 

.98092 

.98249 

.98399 

.98543 

.98679 

.98809 

\98932 

.99048 

20 

9.98094 

9.98251 

9.98402 

9.98545 

9.98681 

9.98811 

9.98934 

9.99050 

24 

.98097 

.98254 

.98404 

.98547 

.98684 

.98813 

.98936 

.99052 

28 

.98100 

.98256 

.98406 

.98550 

.98686 

.98815 

.98938 

.99054 

32 

.98102 

.98259 

.98409 

.98552 

.98688 

.98817 

.98940 

.99056 

36 

.98105 

.98262 

.98411 

.98554 

.98690 

.98819 

.98942 

.99058 

40 

9.98108 

9.98364 

9.98414 

9.98557 

9.98692 

9.98822 

9.98944 

9.99059 

44 

.98110 

.98267 

.98416 

.98559 

.98695 

.98824 

.98946 

.99061 

48 

.98113 

.98269 

.98419 

.98561 

.98697 

.98826 

.98948 

.99063 

s* 

.98116 

.98272 

.98421 

.98564 

.98699 

.98828 

.98950 

.99065 

56 

.98118 

.98274 

.98424 

.98566 

.98701 

.98830 

.98952 

.99067 

8 

10h  25m  10h  29m 

10*  33m  10*  37m 

10*  41  "l  10*  45m 

10*  49m  10*  53m 

0 

9.98121 

9.98277 

9.98426 

9.98568 

9.98703 

9.98832 

9.98954 

9.99069 

4 

.98124 

.98279 

.98428 

.98570 

.98706 

.98834 

.98956 

.99071 

8 

.98126 

.98282 

.98431 

.98573 

.98708 

.98836 

.98958 

.99072 

12 

.98129 

.98285 

.98433 

.98575 

.98710 

.98838 

.98960 

.99074 

16 

.98132 

.98287 

.98436 

.98577 

.98712 

.98840 

.98962 

.99076 

20 

9.98134 

9.98290 

9.98438 

9.98580 

9.98714 

9.98842 

9.98964 

9.99078 

24 

.98137 

.98292 

.98440 

.98582 

.98717 

.98845 

.98966 

.99080 

28 

.98139 

.98295 

.98443 

.98584 

.98719 

.98847 

.98968 

.99082 

32 

.98142 

.98297 

.98445 

.98587 

.98721 

.98849 

.98970 

.99084 

36 

.98145 

.98300 

.98448 

.98589 

.98723 

.98851 

.98971 

.99085 

40 

9.98147 

9.98302 

9.98450 

9.98591 

9.98725 

9.98853 

9.98973 

9.99087 

44 

.98150 

.98305 

.98453 

.98593 

.98728 

.98855 

.98975 

.99089 

48 

.98153 

.98307 

.98455 

.98596 

.98730 

.98857 

.98977 

.99091 

52 

.98155 

.98310 

.98457 

.98598 

.98732 

.98859 

.98979 

.99093 

56 

.98158 

.98312 

.98460 

.98600 

.98734 

.98861 

.98981 

.99095 

s 

10*  26m  10*  30m 

10*  34m  10*  38™ 

10*  42m  10*  46m 

10*  50m  10*  54m 

0 

9.98161 

9.98315 

9.98462 

9.98603 

9.98736 

9.98863 

9.98983 

9.99096 

4 

.98163 

.98317 

.98465 

.98605 

.98738 

.98865 

.98985 

.99098 

8 

.98166 

.98320 

.98467 

.98607 

.98741 

.98867 

.98987 

.99100 

12 

.98168 

.98322 

.98469 

.98609 

.98743 

.98869 

.98989 

.99102 

16 

.98171 

.98325 

.98472 

.98612 

.98745 

.98871 

.98991 

.99104 

20 

9.98174 

9.98327 

9.98474 

9.98614 

9.98747 

9.98873 

9.98993 

9.99106 

24 

.98176 

.98330 

.98476 

.98616 

.98749 

.98875 

.98995 

.99107 

28 

.98179 

.98332 

.98479 

.98619 

.98751 

.98877 

.98997 

.99109 

32 

.98182 

.98335 

.98481 

.98621 

.98754 

.98880 

.98999 

.99111 

36 

.98184 

.98337 

.98484 

.98623 

.98756 

.98882 

.99001 

.99113 

40 

9.98187 

9.98340 

9.98486 

9.98625 

9.98758 

9.98884 

9.99003 

9.99115 

44 

.98189 

.98342 

.98488 

.98628 

.98760 

.98886 

.99004 

.99116 

48 

.98192 

.98345 

.98491 

.98630 

.98762 

.98888 

.99006 

.99118 

52 

.98195 

.98347 

.98493 

.98632 

.98764 

.98890 

.99008 

.99120 

56 

.98197 

.98350 

.98496 

.98634 

.98766 

.98892 

.99010 

.99122 

s 

10*27m  10*31™ 

10*  35m  10*  39m 

10*  43m  10*  47m 

10*51m  10*  55m 

0 

9.98200 

9.98352 

9.98498 

9.98637 

9.98769 

9.98894 

9.99012 

9.99124 

4 

.98202 

.98355 

.98500 

.98639 

.98771 

.98896 

.99014 

.99126 

8 

.98205 

.98357 

.98503 

.98641 

.98773 

.98898 

.99016 

.99127 

12 

.98208 

.98360 

.98505 

.98643 

.98775 

.98900 

.99018 

.99129 

16 

.98210 

.98362 

.98507 

.98646 

.98777 

.98902 

.99020 

.99131 

20 

9.98213 

9.98365 

9.98510 

9.98648 

9.98779 

9.98904 

9.99022 

9.99133 

24 

.98215 

.98367 

.98512 

.98650 

.98781 

.98906 

.99024 

.99135 

28 

.98218 

.98370 

.98514 

.98652 

.98784 

.98908 

.99026 

.99136 

32 

.98221 

.98372 

.98517 

.98655 

.98786 

.98910 

.99027 

.99138 

36 

.98223 

.98375 

.98519 

.98657 

.98788 

.98912 

.99029 

.99140 

40 

9.98226 

9.98377 

9.98521 

9.98659 

9.98790 

9.98914 

9.99031 

9.99142 

44 

.98228 

.98379 

.98524 

.98661 

.98792 

.98916 

.99033 

.99143 

48 

.98231 

.98382 

.98526 

.98664 

.98794 

.98918 

.99035 

.99145 

62 

.98233 

.98384 

.98529 

.98666 

.98796 

.98920 

.99037 

.99147 

56 

.98236 

.98387 

.98531 

.98668 

.98798 

.98922 

.99039 

.99149 

60 

9.98239 

9.98389 

9.98533 

9.98670 

9.98801 

9.98924 

9.99041 

9.99151 

282 


Table  10.    Haversine  Table 


s 

10*  56™  Ilh0m 

llh  jrn   llh  #m 

11*  12m  llh  Iftn 

Uh  20m  llh  24m 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.99151 

9.99254 

9.99350 

9.99440 

9.99523 

9.99599 

9.99669 

9.99732 

4 

.99152 

.99255 

.99352 

.99441 

.99524 

.99600 

.99670 

.99733 

8 

.99154 

.99257 

.99353 

.99443 

.99526 

.99602 

.99671 

.99734 

12 

.99156 

.99259 

.99355 

.99444 

.99527 

.99603 

.99672 

.99735 

16 

.99158 

.99260 

.99356 

.99446 

.99528 

.99604 

.99673 

.99736 

20 

9.99159 

9.99262 

9.99358 

9.99447 

9.99529 

9.99605 

9.99674 

9.99737 

24 

.99161 

.99264 

.99359 

.99448 

.99531 

.99606 

.99675 

.99738 

28 

.99163 

.99265 

.99361 

.99450 

.99532 

.99608 

.99677 

.99739 

32 

.99165 

.99267 

.99362 

.99451 

.99533 

.99609 

.99678 

.99740 

36 

.99166 

.99269 

.99364 

.99453 

.99535 

.99610 

.99679 

.99741 

40 

9.99168 

9.99270 

9.99366 

9.99454 

9.99536 

9.99611 

9.99680 

9.99742 

44 

.99170 

.99272 

.99367 

.99456 

.99537 

.99612 

.99681 

.99743 

48 

.99172 

.99274 

.99369 

.99457 

.99539 

.99614 

.99682 

.99744 

52 

.99173 

.99275 

.99370 

.99458 

.99540 

.99615 

.99683 

.99745 

56 

.99175 

.99277 

.99372 

.99460 

.99541 

.99616 

.99684 

.99746 

s 

IQh  Qfm  llh  im 

11*  5m  11^  9^ 

Hh  igm  llh  17m 

Ilh21m  Ilh25m 

0 

9.99177 

9.99278 

9.99373 

9.99461 

9.99543 

9.99617 

9.99685 

9.99747 

4 

.99179 

.99280 

.99375 

.99463 

.99544 

.99618 

.99686 

.99748 

8 

.99180 

.99282 

.99376 

.99464 

.99545 

.99620 

.99687 

.99748 

12 

.99182 

.99283 

.99378 

.99465 

.99546 

.99621 

.99688 

.99749 

16 

.99184 

.99285 

.99379 

.99467 

.99548 

.99622 

.99690 

.99750 

20 

9.99186 

9.99287 

9.99381 

9.99468 

9.99549 

9.99623 

9.99691 

9.99751 

24 

.99187 

.99288 

.99382 

.99470 

.99550 

.99624 

.99692 

.99752 

28 

.99189 

.99290 

.99384 

.99471 

.99552 

.99626 

.99693 

.99753 

32 

.99191 

.99291 

.99385 

.99472 

.99553 

.99627 

.99694 

.99754 

36 

.99193 

.99293 

.99387 

.99474 

.99554 

.99628 

.99695 

.99755 

40 

9.99194 

9.99295 

9.99388 

9.99475 

9.99555 

9.99629 

9.99696 

9.99756 

44 

.99196 

.99296 

.99390 

.99477 

.99557 

.99630 

.99697 

.99757 

48 

.99198 

.99298 

.99391 

.99478 

.99558 

.99631 

.99698 

.99758 

52 

.99200 

.99300 

.99393 

.99479 

.99559 

.99633 

.99699 

.99759 

56 

.99201 

.99301 

.99394 

.99481 

.99561 

.99634 

.99700 

.99760 

s 

IQh  58™  llh  2™ 

Hh  Qm  nh  i0m 

llh  Uf*  nh  I8m 

llh  22m  Uh  26™ 

0 

9.99203 

9.99303 

9.99396 

9.99482 

9.99562 

9.99635 

9.99701 

9.99761 

4 

.99205 

.99304 

.99397 

.99484 

.99563 

.99636 

.99702 

.99762 

8 

.99206 

.99306 

.99399 

.99485 

.99564 

.99637 

.99703 

.99763 

12 

.99208 

.99308 

.99400 

.99486 

.99566 

.99638 

.99704 

.99764 

16 

.99210 

.99309 

.99402 

.99488 

.99567 

.99639 

.99705 

.99765 

20 

9.99212 

9.99311 

9.99403 

9.99489 

9.99568 

9.99641 

9.99706 

9.99766 

24 

.99213 

.99312 

.99405 

.99490 

.99569 

.99642 

.99707 

.99766 

28  ' 

.99215 

.99314 

.99406 

.99492 

.99571 

.99643 

.99708 

.99767 

32 

.99217 

.99316 

.99408 

.99493 

.99572 

.99644 

.99710 

.99768 

36 

.99218 

.99317 

.99409 

.99495 

.99573 

.99645 

.99711 

.99769 

40 

9.99220 

9.99319 

9.99411 

9.99496 

9.99575 

9.99646 

9.99712 

9.99770 

44 

.99222 

.99320 

.99412 

.99497 

.99576 

.99648 

.99713 

.99771 

48 

.99223 

.99322 

.99414 

.99499 

.99577 

.99649 

.99714 

.99772 

52 

.99225 

.99324 

.99415 

.99500 

.99578 

.99650 

.99715 

.99773 

56 

.99227 

.99325 

.99417 

.99501 

.99580 

.99651 

.99716 

.99774 

s 

10h  59m  11^  sm 

Hh  7™  llh  jjm 

Hh  IQm  Uh  igm 

llh  23m  nh  27m 

0 

9.99229 

9.99327 

9.99418 

9.99503 

9.99581 

9.99652 

9.99717 

9.99774 

4 

.99230 

.99328 

.99420 

.99504 

.99582 

.99653 

.99718 

.99775 

8 

.99232 

.99330 

.99421 

.99505 

.99583 

.99654 

.99719 

.99776 

12 

.99234 

.99331 

.99422 

.99507 

.99584 

.99655 

.99720 

.99777 

16 

.99235 

.99333 

.99424 

.99508 

.99586 

.99657 

.99721 

.99778 

20 

9.99237 

9.99335 

9.99425 

9.99510 

9.99587 

9.99658 

9.99722 

9.99779 

24 

.99239 

.99336 

.99427 

.99511 

.99588 

.99659 

.99723 

.99780 

28 

.99240 

.99338 

.99429 

.99512 

.99589 

.99660 

.99724 

.99781 

32 

.99242 

.99339 

.99430 

.99514 

.99591 

.99661 

.99725 

.99782 

36 

.99244 

.99341 

.99431 

.99515 

.99592 

.99662 

.99726 

.99783 

40 

9.99245 

9.99342 

9.99433 

9.99516 

9.99593 

9.99663 

9.99727 

9.99784 

44 

.99247 

.99344 

.99434 

.99518 

.99594 

.99664 

.99728 

.99785 

48 

.99249 

.99345 

.99436 

.99519 

.99596 

.99666 

.99729 

.99786 

52 

.99250 

.99347 

.99437 

.99520 

.99597 

.99667 

.99730 

.99786 

56 

.99252 

.99349 

.99438 

.99522 

.99598 

.99668 

.99731 

.99787 

60 

9.99254 

9.99350 

9.99440 

9.99523 

9.99599 

9.99669 

9.99732 

9.99788 

Table  10.    Haversine  Table 


283 


s 

11*  28™  llh  32™ 

11*  36™  11*  40m 

11*  Um  Hh  48™ 

ll*52m  11*56™ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.99788 

9.99S3S 

9.99881 

9.99917 

9.99947 

9.99970 

9.99987 

9.99997 

4 

.99789 

.99839 

.99882 

.99918 

.99948 

.99971 

.99987 

.99997 

8 

.99790 

.99839 

.99882 

.99918 

.99948 

.99971 

.99987 

.99997 

12 

.99791 

.99840 

.99883 

.99919 

.99948 

.99971 

.99987 

.99997 

16 

.99792 

.99841 

.99884 

.99919 

.99949 

.99972 

.99988 

.99997 

20 

9.99793 

9.99842 

9.99884 

9.99920 

9.99949 

9.99972 

9.99988 

9.99997 

24 

.99793 

.99842 

.99885 

.99921 

.99950 

.99972 

.99988 

.99997 

28 

.99794 

.99843 

.99885 

.99921 

.99950 

.99973 

.99988 

.99997 

32 

.99795 

.99844 

.99886 

.99922 

.99951 

.99973 

.99988 

.99998 

36 

.99796 

.99845 

.99887 

.99922 

.99951 

.99973 

.99989 

.99998 

40 

9.99797 

9.99845 

9.99887 

9.99923 

9.99951 

9.99973 

9.99989 

9.99998 

44 

.99798 

.99846 

.99888 

.99923 

.99952 

.99974 

.99989 

.99998 

48 

.99799 

.99847 

.99889 

.99924 

.99952 

.99974 

.99989 

.99998 

52 

.99800 

.99848 

.99889 

.99924 

.99953 

.99974 

.99989 

.99998 

56 

.99800 

.99848 

.99890 

.99925 

.99953 

.99975 

.99990 

.99998 

s 

llh  29m  11*  33m 

11*  37m  11*  41m 

11*  45m  11*  49m 

11*  53™  11*  57™ 

0 

9.99801 

9.99849 

9.99891 

9.99925 

9.99953 

9.99975 

9.99990 

9.99998 

4 

.99802 

.99850 

.99891 

.99926 

.99954 

.99975 

.99990 

.99998 

8 

.99803 

.99851 

.99892 

.99926 

.99954 

.99976 

.99990 

.99998 

12 

.99804 

.99851 

.99893 

.99927 

.99954 

.99976 

.99990 

.99998 

16 

.99805 

.99852 

.99893 

.99927 

.99955 

.99976 

.99991 

.99998 

20 

9.99805 

9.99853 

9.99894 

9.99928 

9.99955 

9.99976 

9.99991 

9.99999 

24 

.99806 

.99854 

.99894 

.99928 

.99956 

.99977 

.99991 

.99999 

28 

.99807 

.99854 

.99895 

.99929 

.99956 

.99977 

.99991 

.99999 

32 

.99808 

.99855 

.99896 

.99929 

.99957 

.99977 

.99991 

.99999 

36 

.99809 

.99856 

.99896 

.99930 

.99957 

.99978 

.99992 

.99999 

40 

9.99810 

9.99857 

9.99897 

9.99931 

9.99958 

9.99978 

9.99992 

9.99999 

44 

.99811 

.99857 

.99897 

.99931 

.99958 

.99978 

.99992 

.99999 

48 

.99811 

,99858 

.99898 

.99932 

.99958 

.99978 

.99992 

.99999 

52 

.99812 

.99859 

.99899 

.99932 

.99959 

.99979 

.99992 

.99999 

66 

.99813 

.99859 

.99899 

.99933 

.99959 

.99979 

.99992 

.99999 

8 

llh  SO™  llh  $4™ 

11*  38m  11*  42™ 

11*  46m  11*50™ 

11*  54m  11*  58m 

0 

9.99814 

9.99860 

9.99900 

9.99933 

9.99959 

9.99979 

9.99993 

9.99999 

4 

.99815 

.99861 

.99901 

.99934 

.99960 

.99980 

.99993 

.99999 

8 

.99815 

.99862 

.99901 

.99934 

.99960 

.99980 

.99993 

.99999 

12 

.99816 

.99862 

.99902 

.99935 

.99961 

.99980 

.99993 

.99999 

16 

.99817 

.99863 

.99902 

.99935 

.99961 

.99980 

.99993 

.99999 

20 

9.99818 

9.99864 

9.99903 

9.99935 

9.99961 

9.99981 

9.99993 

9.99999 

24 

.99819 

.99864 

.99904 

.99936 

.99962 

.99981 

.99994 

.99999 

28 

.99820 

.99865 

.99904 

.99936 

.99962 

.99981 

.99994 

.00000 

32 

.99820 

.99866 

.99905 

.99937 

.99963 

.99981 

.99994 

.00000 

36 

.99821 

.99867 

.99905 

.99937 

.99963 

.99982 

.99994 

.00000 

40 

9.99822 

9.99867 

9.99906 

9.99938 

9.99963 

9.99982 

9.99994 

0.00000 

44 

.99823 

.99868 

.99906 

.99938 

.99964 

.99982 

.99994 

.00000 

48 

.99824 

.99869 

.99907 

.99939 

.99964 

.99983 

.99994 

.00000 

52 

.99824 

.99869 

.99908 

.99939 

.99964 

.99983 

.99995 

.00000 

56 

.99825 

.99870 

.99908 

.99940 

.99965 

.99983 

.99995 

.00000 

s 

llh  sim  nhsgm 

11*39™  11*  43m 

11*  47m  11*  51m 

11*  55™  11*  5yn 

0 

9.99826 

9.99871 

9.99909 

9.99940 

9.99965 

9.99983 

9.99995 

0.00000 

4 

.99827 

.99871 

.99909 

.99941 

.99965 

.99983 

.99995 

.00000 

8 

.99828 

.99872 

.99910 

.99941 

.99966 

.99984 

.99995 

.00000 

12 

.99828 

.99873 

.99911 

.99942 

.99966 

.99984 

.99995 

.00000 

16 

.99829 

.99874 

.99911 

.99942 

.99966 

.99984 

.99995 

.00000 

20 

9.99830 

9.99874 

9.99912 

9.99943 

9.99967 

9.99984 

9.99996 

0.00000 

24 

.99831 

.99875 

.99912 

.99943 

.99967 

.99985 

.99996 

.00000 

28 

.99832 

.99876 

.99913 

.99943 

.99968 

.99985 

.99996 

.00000 

32 

.99832 

.99876 

.99913 

.99944 

.99968 

.99985 

.99996 

.00000 

36 

.99833 

.99877 

.99914 

.99944 

.99968 

.99985 

.99996 

.00000 

40 

9.99834 

9.99878 

9.99915 

9.99945 

9.99969 

9.99986 

9.99996 

0.00000 

44 

.99835 

.99878 

.99915 

.99945 

.99969 

.99986 

.99996 

.00000 

48 

.99836 

.99879 

.99916 

.99946 

.99969 

.99986 

.99996 

.00000 

52 

.99836 

.99880 

.99916 

.99946 

.99970 

.99986 

.99996 

.00000 

56 

.99837 

.99880 

.99917 

.99947 

.99970 

.99987 

.99997 

.00000 

60 

9.99838 

9.99881 

9.99917 

9.99947 

9.99970 

9.99987 

9.99997 

0.00000 

284 


Table  11.    Azimuth 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

12h  0m 

12h  sm 

12*16™ 

12h  24m 

12*32™ 

12h  40m 

12*  48m 

12*  56m 

THESE 

IN   —  >• 

^    i  \ 

FORE- 

24  0 

28  52 

23  44 

23  36 

23  28 

23  20 

23  12 

23  4 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

Oh  Qm 

Qh  gm 

Oh16m 

0*24™ 

QhS2m 

0*40™ 

0*48™ 

0*56™ 

THESE 

IN   —  >• 

<      IN 

AFTER- 

12 0 

n  52 

11  44 

11  36 

11  28 

11  20 

11  12 

11   4 

AFTER- 

NOON 

NOON 

0° 

0 

349 

698 

1045 

1392 

1737 

2079 

2419 

Oo 

2 

0 

349 

697 

1045 

1391 

1736 

2078 

2417 

2 

4 

0 

348 

696 

1042 

1389 

1732 

2074 

2413 

4 

6 

0 

347 

694 

1040 

1384 

1726 

2067 

2406 

6 

8 

0 

346 

691 

1035 

1378 

1720 

2059 

2395 

8 

10 

0 

344 

687 

1029 

1371 

1710 

2047 

2382 

10 

12 

0 

341 

682 

1022 

1361 

1698 

2033 

2367 

12 

14 

0 

339 

677 

1015 

1351 

1685 

2018 

2347 

14 

16 

0 

336 

671 

1005 

1338 

1669 

1998 

2326 

16 

18 

0 

332 

663 

994 

1323 

1651 

1977 

2301 

18 

20 

0 

328 

656 

982 

1308 

1632 

1954 

2274 

20 

22 

0 

324 

647 

969 

1290 

1610 

1928 

2244 

22 

24 

0 

319 

637 

955 

1272 

1586 

1900 

2210 

24 

26 

0 

314 

627 

940 

1251 

1561 

1868 

2174 

26 

28 

0 

308 

616 

923 

1228 

1533 

1835 

2136 

28 

30 

0 

302 

604 

905 

1205 

1504 

1801 

2095 

30 

32 

0 

296 

592 

886 

1180 

1472 

1763 

2051 

32 

34 

0 

289 

578 

867 

1153 

1440 

1724 

2005 

34 

36 

0 

282 

564 

846 

1126 

1405 

1682 

1957 

36 

01 

38 

0 

275 

550 

824 

1096 

1369 

1639 

1906 

38 

X 

o 

40 

0 

267 

534 

801 

1066 

1330 

1592 

1853 

40 

03 
W 

g 

42 

0 

259 

518 

777 

1034 

1290 

1545 

1798 

42 

Q 
p 

• 

44 

0 

251 

502 

752 

1001 

1249 

1496 

1740 

44 

H 

1 

46 

0 

242 

485 

726 

967 

1206 

1444 

1681 

46 

i 

m 

48 

0 

234 

467 

699 

931 

1162 

1391 

1619 

48 

<j 

Q 

50 

0 

224 

448 

672 

895 

1116 

1337 

1555 

50 

52 

0 

215 

429 

644 

857 

1069 

1280 

1489 

52 

54 

0 

205 

411 

615 

818 

1021 

1222 

1422 

54 

56 

0 

195 

390 

585 

778 

971 

1162 

1353 

56 

58 

0 

185 

370 

554 

738 

920 

1102 

1282 

58 

60 

0 

175 

349 

523 

696 

868 

1040 

1210 

60 

62 

0 

164 

328 

490 

653 

815 

976 

1136 

62 

64 

0 

153 

306 

458 

610 

761 

911 

1060 

64 

66 

0 

142 

284 

425 

566 

706 

846 

984 

66 

68 

0 

131 

261 

392 

521 

651 

779 

906 

68 

70 

0 

119 

239 

358 

476 

594 

711 

827 

70 

72 

0 

108 

216 

323 

430 

537 

643 

748 

72 

74 

0 

96 

192 

288 

384 

479 

573 

667 

74 

76 

0 

84 

169 

253 

337 

420 

503 

585 

76 

78 

0 

73 

145 

217 

289 

361 

432 

503 

78 

80 

0 

61 

121 

182 

242 

302 

361 

420 

80 

82 

0 

49 

97 

146 

194 

242 

289 

337 

82 

84 

0 

36 

73 

109 

146 

182 

217 

253 

84 

86 

0 

24 

49 

73 

97 

121 

145 

169 

86 

88 

0 

12 

24 

36 

49 

61 

73 

84 

88 

USE 

USE 

THESE 

0° 

2° 

4° 

6° 

8° 

10° 

12° 

14° 

THESE 

IN      ^ 

•<  —   IN 

FORE- 

180 

178 

176 

174 

172 

170 

168 

166 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

180° 

182° 

184° 

186° 

188° 

190° 

192° 

194° 

THESE 

IN     ^ 

-<  —    IN 

AFTER- 

360 

358 

356 

354 

352 

350 

348 

346 

AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

Table  11.    Azimuth 


285 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

13*  4m 

13*12™ 

13*  20™ 

13*28™ 

13*36™ 

13*  44™ 

13*  52™ 

THESE 

IN    —  >• 

FORE- 

22 66 

22  48 

22  40 

22  32 

22  24 

22  16 

22  8 

•<  —   IN 
FORE- 

NOON 

NOON 

USE 

USE 

THESE 

Jh  jrn 

1*12™ 

l*20m 

1*28™ 

1*36™ 

l*44m 

1*52™ 

THESE 

IN     ^ 

•<  —    IN 

AFTER- 

10 56 

10  48 

10  40 

10  32 

10  24 

10  16 

10  8 

AFTER- 

NOON 

NOON 

0° 

2756 

3090 

3421 

3746 

4067 

4383 

4695 

0° 

2 

2755 

3088 

3418 

3744 

4065 

4381 

4692 

2 

4 

2750 

3082 

3412 

3737 

4058 

4373 

4684 

4 

6 

2742 

3073 

3402 

3726 

4044 

4360 

4669 

6 

8 

2730 

3060 

3387 

3710 

4028 

4341 

4649 

8 

10 

2714 

3043 

3368 

3689 

4005 

4317 

4624 

10 

12 

2696 

3023 

3346 

3664 

3978 

4287 

4592 

12 

14 

2674 

2998 

3319 

3635 

3947 

4253 

4555 

14 

16 

2650 

2970 

3288 

3600 

3910 

4214 

4513 

16 

18 

2622 

2939 

3253 

3563 

3868 

4170 

4465 

18 

20 

2590 

2903 

3214 

3521 

3821 

4119 

4412 

20 

22 

2556 

2865 

3171 

3473 

3771 

4064 

4353 

22 

24 

2519 

2823 

3125 

3422 

3715 

4005 

4288 

24 

26 

2477 

2777 

3074 

3367 

3656 

3941 

4220 

26 

28 

2434 

2729 

3020 

3308 

3591 

3871 

4145 

28 

30 

2387 

2676 

2962 

3244 

3522 

3796 

4065 

30 

32 

2337 

2620 

2900 

3177 

3449 

3718 

3981 

32 

34 

2285 

2563 

2835 

3106 

3372 

3634 

3892 

34 

36 

2230 

2500 

2767 

3031 

3291 

3546 

3798 

36 

00 

38 

2172 

2435 

2696 

2952 

3205 

3454 

3699 

38 

K 
O 

40 

2112 

2367 

2620 

2869 

3116 

3358 

3596 

40 

» 

H   42 

2048 

2297 

2542 

2784 

3023 

3258 

3489 

42 

Q 

£   44 

1983 

2223 

2460 

2695 

2925 

3154 

3377 

44 

i 

46 

1914 

2147 

2375 

2602 

2826 

3045 

3261 

46 

§ 

a   48 

1845 

2067 

2289 

2507 

2721 

2934 

3142 

48 

<3 

Q 

50 

1772 

1986 

2199 

2408 

2614 

2818 

3018 

50 

52 

1697 

1902 

2106 

2306 

2504 

2699 

2891 

52 

54 

1620 

1818 

2010 

2202 

2391 

2577 

2759 

54 

56 

1541 

1728 

1913 

2094 

2275 

2451 

2625 

56 

58 

1461 

1638 

1812 

1986 

2155 

2324 

2488 

58 

60 

1378 

1545 

1710 

1874 

2033 

2192 

2347 

60 

62 

1294 

1451 

1606 

1759 

1909 

2058 

2204 

62 

64 

1209 

1355 

1499 

1643 

1783 

1922 

2058 

64 

66 

1121 

1257 

1391 

1524 

1654 

1783 

1909 

66 

68 

1032 

1158 

1281 

1404 

1524 

1643 

1759 

68 

70 

943 

1057 

1169 

1281 

1391 

1499 

1606 

70 

72 

852 

955 

1057 

1158 

1257 

1355 

1451 

72 

74 

760 

852 

943 

1032 

1121 

1209 

1294 

74 

76 

667 

748 

827 

906 

984 

1060 

1136 

76 

78 

573 

643 

711 

779 

846 

911 

976 

78 

80 

479 

537 

594 

651 

706 

761 

815 

80 

82 

384 

430 

476 

521 

566 

610 

653 

82 

84 

288 

323 

358 

392 

425 

458 

491 

84 

86 

192 

216 

239 

261 

290 

306 

328 

86 

88 

96 

108 

119 

131 

142 

153 

164 

88 

USE 

USE 

THESE 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

THESE 

IN   —  >• 

^      IN 

FORE- 

164 

162 

160 

158 

156 

154 

152 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

196° 

198° 

200° 

202° 

204° 

206° 

208° 

THESE 

IN   —  >• 

^      IN 

AFTER- 

344 

342 

340 

338 

336 

334 

332 

AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

286 


Table  11.    Azimuth 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

14*  Om 

14*  8™ 

Up  16™ 

14*S4m 

14*32™ 

14h40m 

14*48" 

14*56™ 

THESE 

IN   —  >• 

•<  —   IN 

FORE- 

22 0 

21  52 

21  44 

21  36 

21  28 

21  20 

21  12 

21  4 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

2h  Om 

2h  s™ 

2*16™ 

2*24™ 

2*32™ 

2*40™ 

2*48™ 

2*56™ 

THESE 

IN      ^ 

^      IN 

AFTER- 

10  0 

9  52 

9  44 

9  36 

9  28 

9  20 

9  12 

9  4 

AFTER- 

NOON 

NOON 

0° 

5000 

5299 

5593 

5878 

6156 

6428 

6691 

6947 

Qo 

2 

4997 

5297 

5589 

5875 

6153 

6424 

6688 

6942 

2 

4 

4987 

5286 

5578 

5864 

6142 

6412 

6676 

6929 

4 

6 

4973 

5270 

5562 

5845 

6124 

6393 

6655 

6908 

6 

8 

4951 

5248 

5538 

5821 

6096 

6365 

6627 

6879 

8 

10 

4923 

5219 

5507 

5789 

6063 

6330 

6591 

6841 

10 

12 

4891 

5183 

5470 

5749 

6022 

6288 

6545 

6795 

12 

14 

4852 

5141 

5426 

5703 

5973 

6237 

6492 

6741 

14 

16 

4806 

5094 

5375 

5650 

5919 

6179 

6433 

6677 

16 

18 

4755 

5040 

5319 

5590 

5856 

6113 

6363 

6607 

18 

20 

4699 

4979 

5255 

5524 

5785 

6040 

6288 

6528 

20 

22 

4635 

4914 

5184 

5450 

5709 

5960 

6204 

6440 

22 

24 

4567 

4841 

5109 

5370 

5624 

5872 

6112 

6346 

24 

26 

4493 

4763 

5025 

5283 

5534 

5777 

6015 

6243 

26 

28 

4415 

4678 

4938 

5190 

5437 

5675 

5907 

6134 

28 

30 

4330 

4588 

4843 

5091 

5332 

5567 

5794 

6016 

30 

32 

4240 

4493 

4742 

4984 

5222 

5451 

5674 

5891 

32 

34' 

4145 

4393 

4635 

4873 

5104 

5328 

5547 

5758 

34 

36 

4044 

4287 

4524 

4755 

4980 

5200 

5414 

5619 

36 

w 

38 

3941 

4176 

4407 

4633 

4852 

5065 

5272 

5474 

38 

* 

o 

40 

3830 

4060 

4284 

4503 

4717 

4923 

5126 

5321 

40 

1 

1 

42 

3715 

3939 

4156 

4368 

4575 

4776 

4973 

5162 

42 

Q 

I 

44 

3596 

3812 

4023 

4229 

4429 

4624 

4812 

4997 

44 

E 

3 

46 

3473 

3681 

3885 

4083 

4277 

4465 

4648 

4825 

46 

1 

§ 

48 

3346 

3546 

3742 

3932 

4120 

4301 

4477 

4648 

48 

•3 

Q 

50 

3214 

3406 

3594 

3779 

3958 

4131 

4301 

4465 

50 

52 

3078 

3263 

3443 

3619 

3790 

3958 

4120 

4277 

52 

54 

2939 

3115 

3287 

3454 

3619 

3779 

3932 

4083 

54 

56 

2796 

2963 

3127 

3287 

3443 

3594 

3742 

3885 

56 

58 

2650 

2808 

2963 

3115 

3263 

3406 

3546 

3681 

58 

60 

2500 

2650 

2796 

2939 

3078 

3214 

3346 

3473 

60 

62 

2347 

2488 

2625 

2760 

2891 

3018 

3142 

3261 

62 

64 

2192 

2324 

2451 

2577 

2699 

2818 

2934 

3045 

64 

66 

2033 

2155 

2275 

2391 

2504 

2614 

2721 

2826 

66 

68 

1874 

1986 

2094 

2202 

2306 

2408 

2507 

2602 

68 

70 

1710 

1812 

1913 

2010 

2106 

2199 

2289 

2375 

70 

72 

1545 

1638 

1728 

1817 

1902 

1986 

2067 

2147 

72 

74 

1378 

1461 

1541 

1620 

1697 

1720 

1845 

1914 

74 

76 

1210 

1282 

1353 

1422 

1489 

1555 

1619 

1681 

76 

78 

1040 

1102 

1162 

1222 

1280 

1337 

1391 

1444 

78 

80 

868 

920 

971 

1021 

1069 

1116 

1162 

1206 

80 

82 

696 

738 

778 

818 

857 

895 

931 

967 

82 

84 

523 

554 

585 

615 

644 

672 

699 

726 

84 

86 

349 

370 

390 

411 

429 

448 

467 

485 

86 

88 

175 

185 

195 

205 

215 

224 

234 

242 

88 

USE 

USE 

THESE 

30° 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

THESE 

IN      ^ 

•^  —    IN 

FORE- 

150 

148 

146 

144 

142 

140 

138 

136 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

210° 

212° 

214° 

216° 

218° 

220° 

222° 

224° 

THESE 

IN     ^ 

•<  —    IN 

AFTER- 

330 

328 

326 

324 

322 

320 

318 

316 

AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

Table  11.    Azimuth 


287 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

15*  4m 

15*  12™ 

15*  20™ 

15*  28m 

15*36™ 

15*  44™ 

15*52™ 

THESE 

IN      ^ 

•<—    IN 

FORE- 

20 56 

20  48 

20  40 

20  32 

20  24 

20  16 

20  8 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

3h  4m 

3*12™ 

3*20™ 

3*28™ 

3*36™ 

3*44m 

3*52™ 

THESE 

IN      ^ 

•<  —    IN 

AFTER- 

8 56 

8  48 

8  40 

8  32 

8  24 

8  16 

8  8 

AFTER- 

NOON 

NOON 

0° 

7193 

7432 

7661 

7879 

8091 

8290 

8480 

0° 

2 

7190 

7427 

7656 

7875 

8085 

8285 

8476 

2 

4 

7176 

7413 

7642 

7861 

8071 

8269 

8461 

4 

6 

7153 

7391 

7619 

7836 

8046 

8245 

8433 

6 

8 

7124 

7358 

7586 

7803 

8011 

8210 

8399 

8 

10 

7084 

7318 

7544 

7761 

7968 

8164 

8352 

10 

12 

7036 

7269 

7494 

7707 

7914 

8110 

8284 

12 

14 

6979 

7211 

7433 

7645 

7850 

8044 

8228 

14 

16 

6915 

7144 

7364 

7575 

7776 

7969 

8153 

16 

18 

6841 

7068 

7286 

7494 

7695 

7884 

8065 

18 

20 

6759 

6984 

7197 

7404 

7603 

7791 

7969 

20 

22 

6670 

6890 

7103 

7307 

7501 

7686 

7863 

22 

24 

6572 

6789 

6998 

7199 

7391 

7573 

7749 

24 

26 

6466 

6679 

6885 

7082 

7271 

7450 

7623 

26 

28 

6352 

6561 

6764 

6958 

7144 

7319 

7489 

28 

30 

6230 

6436 

6634 

6825 

7006 

7180 

7345 

30 

32 

6101 

6302 

6497 

6683 

6861 

7031 

7191 

32 

34 

5964 

6160 

6351 

6533 

6707 

6873 

7031 

34 

36 

5820 

6012 

6197 

6375 

6545 

6707 

6861 

36 

1C 

38 

5669 

5856 

6037 

6210 

6375 

6533 

6683 

38 

fc 

40 

5511 

5693 

5868 

6037 

6197 

6351 

6497 

40 

• 

p 

42 

5346 

5522 

5693 

5856 

6012 

6160 

6302 

42 

1 

•< 
fc 

44 

5175 

5346 

5511 

5669 

5820 

5964 

6101 

44 

^ 

1 

16 

4997 

5162 

5321 

5474 

5619 

5758 

5891 

46 

H 

D 

w 

48 

4812 

4973 

5126 

5272 

5414 

5547 

5674 

48 

< 

Q 

50 

4624 

4776 

4923 

5065 

5200 

5328 

5451 

50 

52 

4429 

4575 

4717 

4852 

4980 

5104 

5222 

52 

54 

4229 

4368 

4503 

4633 

4755 

4873 

4984 

54 

56 

4023 

4156 

4284 

4407 

4524 

4635 

4742 

56 

58 

3812 

3939 

4060 

4176 

4287 

4393 

4493 

58 

60 

3596 

3715 

3830 

3941 

4044 

4145 

4240 

60 

62 

3378 

3489 

3596 

3700 

3798 

3892 

3981 

62 

64 

3154 

3258 

3358 

3454 

3546 

3634 

3718 

64 

66 

2925 

3023 

3116 

3205 

3291 

3372 

3449 

66 

68 

2695 

2784 

2869 

2952 

3031 

3106 

3177 

68 

70 

2460 

2542 

2620 

2696 

2767 

2835 

2900 

70 

72 

2223 

2297 

2367 

2435 

2500 

2563 

2620 

72 

74 

1983 

2048 

2112 

2172 

2230 

2285 

2337 

74 

76 

1740 

1798 

1853 

1906 

1957 

2005 

2051 

76 

78 

1496 

1545 

1592 

1639 

1682 

1724 

1763 

78 

80 

1249 

1290 

1330 

1369 

1405 

1440 

1472 

80 

82 

1001 

1034 

1066 

1096 

1126 

1153 

1180 

82 

84 

752 

777 

801 

824 

846 

867 

886 

84 

86 

502 

518 

534 

550 

564 

578 

592 

86 

88 

251 

259 

267 

275 

282 

289 

296 

88 

USE 

USE 

THESE 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

THESE 

IN    —  >• 

^'•"     Jjq- 

FORE- 

134 

132 

130 

128 

126 

124 

122 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

226° 

228° 

230° 

232° 

234° 

236° 

238° 

THESE 

IN    —  >• 

AFTER- 

314 

312 

310 

308 

306 

304 

302 

•<  —   IN 
AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

288 


Table  11.    Azimuth 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

I  Oh  Qm 

16*  8" 

16*16™ 

16*  24m 

16*32™ 

16*40™ 

16*48™ 

16*36™ 

THESE 

IN      ^ 

FORE- 

20  0 

19  52 

19  44 

19  36 

19  28 

19  20 

19  12 

19  4 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

IN    —  >- 

4*  om 

4*  8^ 

4*  16™ 

4h24m 

4*32m 

4h40" 

4*48™ 

4*36™ 

THESE 

—  -^    ||^ 

AFTER- 

8 0 

7  52 

7  44 

7  36 

7  28 

7  20 

7  12 

7  4 

AFTER- 

NOON 

NOON 

0° 

8660 

8828 

8989 

9135 

9272 

9397 

9510 

9612 

0° 

2 

8656 

8824 

8982 

9131 

9266 

9391 

9506 

9607 

2 

4 

8640 

8808 

8966 

9114 

9249 

9374 

9486 

9590 

4 

6 

8612 

8780 

8939 

9084 

9221 

9346 

9458 

9561 

6 

8 

8576 

8744 

8900 

9046 

9181 

9305 

9419 

9519 

8 

10 

8529 

8696 

8851 

8997 

9131 

9253 

9367 

9466 

10 

12 

8470 

8636 

8792 

8935 

9069 

9191 

9303 

9401 

12 

14 

8403 

8567 

8722 

8863 

8997 

9118 

9228 

3326 

14 

16 

8326 

8487 

8640 

8782 

8913 

9033 

9143 

9241 

16 

18 

8235 

8397 

8549 

8688 

8818 

8937 

9044 

9143 

18 

20 

8137 

8296 

8447 

8584 

8714 

8831 

8937 

9033 

20 

22 

8030 

8187 

8333 

8470 

8596 

8714 

8818 

8913 

22 

24 

7912 

8067 

8212 

8347 

8470 

8584 

8688 

8782 

24 

26 

7784 

7936 

8078 

8212 

8333 

8447 

8549 

8640 

26 

28 

7647 

7796 

7936 

8067 

8187 

8296 

8397 

8487 

28 

30 

7501 

7647 

7784 

7912 

8030 

8137 

8235 

8326 

30 

32 

7345 

7489 

7623 

7749 

7863 

7969 

8065 

8153 

32 

34 

7180 

7319 

7450 

7573 

7686 

7791 

7884 

7969 

34 

36 

7006 

7144 

7271 

7391 

7501 

7603 

7695 

7776 

36 

• 

38 

6825 

6958 

7082 

7199 

7307 

7404 

7494 

7575 

38 

iz; 
0 

40 

6634 

6764 

6885 

6998 

7103 

7197 

7286 

7364 

40 

• 

I 

42 

6436 

6561 

6679 

6789 

6890 

6984 

7068 

7144 

42 

Q 

-<1 
fe 

44 

6230 

6352 

6466 

6572 

6670 

6759 

6841 

6915 

44 

^ 

M 

46 

6016 

6134 

6243 

6346 

6440 

6528 

6607 

6677 

46 

| 

H 

48 

5794 

5907 

6015 

6212 

6204 

6288 

6363 

6433 

48 

< 

Q 

50 

5567 

5675 

5777 

5872 

5960 

6040 

6113 

6179 

50 

52 

5332 

5437 

5534 

5624 

5709 

5785 

5856 

5919 

52 

54 

5091 

5190 

5283 

5370 

5450 

5524 

5590 

5650 

54 

56 

4843 

4938 

5025 

5109 

5184 

5255 

5319 

5375 

56 

58 

4588 

4678 

4763 

4841 

4914 

4979 

5040 

5094 

58 

60 

4330 

4415 

4493 

4567 

4635 

4699 

4755 

4806 

60 

62 

4065 

4145 

4220 

4288. 

4353 

4412 

4465 

4513 

62 

64 

3^96 

3871 

3941 

4005 

4064 

4119 

4170 

4214 

64 

66 

3522 

3591 

3656 

3715 

3771 

3821 

3868 

3910 

66 

68 

3244 

3308 

3367 

3422 

3473 

3521 

3563 

3600 

68 

70 

2962 

3020 

3074 

3125 

3171 

3214 

3253 

3288 

70 

72 

2676 

2729 

2777 

2823 

2865 

2903 

2939 

2970 

72 

74 

2387 

2434 

2477 

2519 

2556 

2590 

2622 

2650 

74 

76 

2095 

2136 

2174 

2210 

2244 

2274 

2301 

2326 

76 

78 

1801 

1835 

1868 

1900 

1928 

1954 

1977 

1998 

78 

80 

1504 

1533 

1561 

1586 

1610 

1632 

1651 

1669 

80 

82 

1205 

1228 

1251 

1272 

1290 

1308 

1323 

1338 

82 

84 

905 

923 

940 

955 

969 

982 

994 

1005 

84 

86 

604 

616 

627 

637 

647 

656 

663 

671 

86 

88 

302 

308 

314 

319 

324 

328 

332 

336 

88 

USE 

USE 

THESE 

60° 

62° 

64° 

66° 

68° 

70° 

72° 

74° 

THESE 

IN    —  >• 

•<  —   IN 

FORE- 

120 

118 

116 

114 

112 

110 

108 

106 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

240° 

242° 

244° 

246° 

248° 

250° 

252° 

254° 

THESE 

IN   —  >• 

^  IN 

AFTER- 

300 

298 

296 

294 

292 

290 

288 

286 

AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

Table  11.    Azimuth 


289 


T,  THE  SHIP'S  APPARENT  TIME  FOR  A  SUN  OBSERVATION, 

OR  THE  HOUR-ANGLE  FOR  A  STAR  OBSERVATION 

USE 

USE 

THESE 

17>>  ^ 

17*12" 

17*20" 

7*28" 

7*36" 

7*44m 

17*52" 

8*  0" 

THESE 

IN   —  ^ 

FORE- 

1856 

18  48 

18  40 

8  32 

8  24 

8  16 

18  8 

8   0 

<  —   IN 
FORE- 

NOON 

NOON 

USE 

USE 

THESE 

5*  4m 

5*12" 

5*20" 

5*28" 

5*36" 

5*44m 

5*52" 

6*  0" 

THESE 

IN    —  > 

AFTER- 

6 56 

6  48 

6  40 

6  32 

6  24 

6  16 

6  8 

6   0 

•<  —    IN 

AFTER- 

NOON 

NOON 

0° 

9703 

9781 

9849 

9904 

9945 

9974 

9993 

10000 

0° 

2 

9696 

9774 

9842 

9897 

9940 

9970 

9988 

9993 

2 

4 

9679 

9757 

9824 

9879 

9922 

9951 

9970 

9974 

4 

6 

9649 

9727 

9795 

9849 

9891 

9922 

9940 

9945 

6 

8 

9610 

9687 

9752 

9806 

9849 

9879 

9897 

9904 

8 

10 

9557 

9634 

9699 

9752 

9795 

9824 

9842 

9849 

10 

12 

9491 

9568 

9634 

9687 

9727 

9757 

9774 

9781 

12 

14 

9414 

9491 

9557 

9610 

9649 

9679 

9696 

9703 

14 

16 

9326 

9401 

9466 

9519 

9561 

9590 

9607 

9612 

16 

18 

9228 

9303 

9367 

9419 

9458 

9486 

9506 

9510 

18 

20 

9118 

9191 

9253 

9305 

9346 

9374 

9391 

9397 

20 

22 

8997 

9069 

9131 

9181 

9221 

9249 

9266 

9272 

22 

24 

8863 

8935 

8997 

9046 

9084 

9114 

9131 

9135 

24 

26 

8722 

8792 

8851 

8900 

8939 

8966 

8982 

8989 

26 

28 

8567 

8636 

8696 

8744 

8780 

8808 

8824 

8828 

28 

30 

8403 

8470 

8529 

8576 

8612 

8640 

8656 

8660 

30 

32 

8228 

8284 

8352 

8399 

8433 

8461 

8476 

8480 

32 

34 

8044 

8110 

8164 

S210 

8245 

8269 

8285 

8290 

34 

36 

7850 

7914 

7968 

8011 

8046 

8071 

8085 

8091 

36 

g 

38 

7645 

7707 

7761 

7803 

7836 

7861 

7875 

7879 

38 

o 

40 

7433 

7494 

7544 

7586 

7619 

7642 

7656 

7661 

40 

to 

• 

§ 

42 

7211 

7269 

7318 

7358 

7391 

7413 

7427 

7432 

42 

8 

i 

44 

6979 

7036 

7084 

7124 

7153 

7176 

7190 

7193 

44 

| 

1 

46 

6741 

6795 

6841 

6879 

6908 

6929 

6942 

6947 

46 

5 

I 

48 

6492 

6545 

6591 

6627 

6655 

6676 

6688 

6691 

48 

<5 

P 

50 

6237 

6288 

6330 

6365 

6393 

6412 

6424 

6428 

50 

52 

5973 

6022 

6063 

6096 

6124 

6142 

6153 

6156 

52 

54 

5703 

5749 

5789 

5821 

5845 

5864 

5875 

5878 

54 

56 

5426 

5470 

5507 

5538 

5562 

5578 

5589 

5593 

56 

58 

5141 

5183 

5219 

5248 

5270 

5286 

5297 

5299 

58 

60 

4852 

4891 

4923 

4951 

4973 

4987 

4997 

5000 

60 

62 

4555 

4592 

4624 

4649 

4669 

4684 

4692 

4695 

62 

64 

4253 

4287 

4317 

4341 

4360 

4373 

4381 

4383 

64 

66 

3947 

3978 

4005 

4028 

4044 

4058 

4065 

4067- 

66 

68 

3635 

3664 

3689 

3710 

3726 

3737 

3744 

3746 

68 

70 

3319 

3346 

3368 

3387 

3402 

3412 

3418 

3421 

70 

72 

2998 

3023 

3043 

3060 

3073 

3082 

3088 

3090 

72 

74 

2674 

2696 

2714 

2730 

2742 

2750 

2755 

2756 

74 

76 

2347 

2367 

2382 

2395 

2406 

2413 

2417 

2419 

76 

78 

2018 

2033 

2047 

2059 

2067 

2074 

2078 

2079 

78 

80 

1685 

1698 

1710 

1720 

1726 

1732 

1736 

1737 

80 

82 

1351 

1361 

1371 

1378 

1384 

1389 

1391 

1392 

82 

84 

1015 

1022 

1029 

1035 

1040 

1042 

1045 

1045 

84 

86 

677 

682 

687 

691 

694 

696 

697 

698 

86 

88 

339 

341 

344 

346 

347 

348 

349 

349 

88 

USE 

USE 

THESE 

76° 

78° 

80° 

82° 

84° 

86° 

88° 

90° 

THESE 

IN   —  >• 

•<  —   IN 

FORE- 

104 

102 

100 

98 

96 

94 

92 

90 

FORE- 

NOON 

NOON 

USE 

USE 

THESE 

256° 

258° 

260° 

262° 

264° 

266° 

268° 

270° 

THESE 

IN    —  >• 

<      IN 

AFTER- 

284 

282 

280 

278 

276 

274 

272 

270 

AFTER- 

NOON 

NOON 

TRUE  BEARING  OR  AZIMUTH 

290 


Table  12.    Auxiliary  Azimuth  Table 


LATITUDE 

DECLINATIONS 

0° 

2° 

4° 

6° 

8° 

10° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

0° 

0° 

2 

0 

90° 

4 

0 

30 

90° 

6 

0 

20 

42 

90° 

8 

0 

15 

30 

49 

90° 

10 

0 

12 

24 

37 

53 

90° 

12 

0 

10 

20 

30 

42 

57 

90° 

14 

0 

8 

17 

26 

35 

46 

59 

90° 

16 

0 

7 

15 

22 

30 

39 

49 

61 

90° 

18  . 

0 

6 

13 

20 

27 

34 

42 

52 

63 

90° 

20 

0 

6 

12 

18 

24 

31 

37 

45 

54 

65 

90° 

22 

0 

5 

11 

16 

22 

28 

34 

40 

47 

56 

66 

90° 

24 

0 

5 

10 

15 

20 

25 

31 

36 

43 

49 

57 

67 

90° 

26 

0 

5 

9 

14 

19 

23 

28 

34 

39 

45 

51 

59 

68 

28 

0 

4 

9 

13 

17 

22 

26 

31 

36 

41 

47 

53 

60 

30 

0 

4 

8 

12 

16 

20 

25 

29 

33 

38 

43 

49 

54 

32 

0 

4 

8 

11 

15 

19 

23 

27 

31 

36 

40 

45 

50 

34 

0 

4 

7 

11 

14 

18 

22 

26 

30 

34 

38 

42 

47 

36 

0 

3 

7 

10 

14 

17 

21 

24 

28 

32 

36 

40 

44 

38 

0 

3 

7 

10 

13 

16 

20 

23 

27 

30 

34 

37 

41 

40 

0 

3 

6 

9 

12 

16 

19 

22 

25 

29 

32 

36 

39 

42 

0 

3 

6 

9 

12 

15 

18 

21 

24 

28 

31 

34 

37 

44 

0 

3 

6 

9 

12 

14 

17 

20 

23 

26 

30 

33 

36 

46 

0 

3 

6 

8 

11 

14 

17 

20 

23 

25 

28 

31 

34 

48 

0 

3 

5 

8 

11 

14 

16 

19 

22 

25 

27 

30 

33 

50 

0 

3 

5 

8 

10 

13 

16 

18 

21 

24 

27 

29 

32 

52 

0 

3 

5 

8 

10 

13 

15 

18 

20 

23 

26 

28 

31 

54 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30 

56 

0 

2 

5 

7 

10 

12 

15 

17 

19 

22 

24 

27 

29 

58 

0 

2 

5 

7 

9 

12 

14 

17 

19 

21 

24 

26 

29 

60 

0 

2 

5 

7 

9 

12 

14 

16 

19 

21 

23 

26 

28 

Table  12.    Completed 


LATITUDE 

DECLINATIONS 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

26° 

90° 

28 

69 

90° 

30 

61 

70 

90° 

32 

56 

62 

71 

90° 

34 

52 

57 

63 

71 

90° 

36 

48 

53 

58 

64 

72 

90° 

38 

45 

50 

54 

59 

65 

73 

90° 

40 

43 

47 

51 

56 

60 

66 

73 

90° 

42 

41 

45 

48 

53 

57 

61 

67 

74 

90° 

44 

39 

43 

46 

50 

54 

58 

62 

68 

74 

90° 

46 

38 

41 

44 

47 

51 

55 

59 

63 

68 

75 

90° 

48 

36 

39 

42 

45 

49 

52 

56 

60 

64 

69 

75 

90° 

50 

35 

38 

41 

44 

47 

50 

53 

57 

61 

65 

70 

76 

90° 

52 

34 

37 

39 

42 

45 

48 

51 

55 

58 

62 

66 

71 

76 

54 

33 

35 

38 

41 

44 

47 

50 

53 

56 

59 

63 

67 

71 

56 

32 

34 

37 

40 

42 

45 

48 

51 

54 

57 

60 

64 

68 

58 

31 

33 

36 

39 

41 

44 

47 

49 

52 

55 

58 

61 

65 

60 

30 

33 

35 

38 

40 

43 

45 

48 

51 

53 

56 

59 

62 

292 


Table  13.    Kelvin's  Simmer  Line  Table 


a  = 

0° 

a  = 

1° 

a  = 

2° 

a  = 

3° 

a  = 

4° 

a  = 

5° 

a  = 

6° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

0  0 

0  0 

1  0 

0  0 

2  0 

0  0 

3  0 

0  0 

4  0 

0  0 

5  0 

0  0 

6  0 

1 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

2 

2  0 

0 

2  0 

0 

2  0 

0 

2  0 

0 

2  0 

0 

2  0 

0 

59 

0 

3 

3  0 

0 

3  0 

0 

3  0 

0 

3  0 

0 

3  0 

0 

59 

0 

259 

0 

4 

4  0 

0 

4  0 

0 

4  0 

0 

4  0 

0 

59 

1 

359 

1 

359 

1 

5 

5  0 

0 

5  0 

0 

5  0 

0 

5  0 

1 

459 

1 

459 

1 

458 

1 

6 

6  0 

0 

6  0 

0 

6  0 

1 

6  0 

1 

559 

1 

559 

2 

558 

2 

7 

7  0 

0 

7  0 

0 

7  0 

1 

59 

1 

659 

2 

658 

2 

658 

3 

8 

8  0 

0 

8  0 

1 

8  0 

1 

759 

2 

759 

2 

758 

3 

757 

4 

9 

9  0 

0 

9  0 

1 

9  0 

1 

859 

2 

859 

3 

858 

4 

857 

5 

10 

10  o- 

0 

10  0 

1 

10  0 

2 

959 

3 

959 

4 

958 

5 

957 

6 

11 

11  0 

0 

11  0 

1 

11  0 

2 

1059 

3 

1058 

4 

1057 

6 

1056 

7 

12 

12  0 

0 

12  0 

1 

12  0 

3 

11  59 

4 

11  58 

5 

11  57 

7 

11  56 

8 

13 

13  0 

0 

13  0 

2 

13  0 

3 

1259 

5 

1258 

6 

1257 

8 

1256 

9 

14 

14  0 

0 

14  0 

2 

59 

4 

1359 

5 

1358 

7 

1357 

9 

1355 

11 

15 

15  0 

0 

15  0 

2 

1459 

4 

1459 

6 

1458 

8 

1456 

10 

1455 

13 

16 

16  0 

0 

16  0 

2 

1559 

5 

1559 

7 

1558 

10 

1556 

12 

1555 

14 

17 

17  0 

0 

17  0 

3 

1659 

5 

1659 

8 

1657 

11 

1656 

14 

1654 

16 

18 

18  0 

0 

18  0 

3 

1759 

6 

1758 

9 

1757 

12 

1756 

15 

1754 

18 

19 

19  0 

0 

19  0 

3 

1859 

7 

1858 

10 

1857 

14 

1855 

17 

1854 

21 

20 

20  0 

0 

20  0 

4 

1959 

8 

1958 

11 

1957 

15 

1955 

19 

1953 

23 

21 

21  0 

0 

21  0 

4 

2059 

9 

2058 

13 

2057 

17 

2055 

21 

2053 

25 

22 

22  0 

0 

22  0 

5 

21  59 

9 

2158 

14 

21  57 

19 

2155 

23 

2152 

28 

23 

23  0 

0 

23  0 

5 

2259 

10 

2258 

15 

2256 

21 

2254 

26 

2252 

31 

24 

24  0 

0 

24  0 

6 

2359 

11 

2358 

17 

2356 

23 

2354 

28 

2352 

34 

25 

25  0 

0 

25  0 

6 

2459 

12 

2458 

18 

2456 

25 

2454 

31 

2451 

37 

26 

26  0 

0 

26  0 

7 

2559 

13 

2558 

20 

2556 

27 

2554 

34 

2551 

40 

27 

27  0 

0 

27  0 

7 

2659 

15 

2658 

22 

2656 

29 

2653 

37 

2650 

44 

28 

28  0 

0 

28  0 

8 

2759 

16 

2757 

24 

2756 

32 

2753 

40 

2750 

47 

29 

29  0 

0 

29  0 

9 

2859 

17 

2857 

26 

2855 

34 

2853 

43 

2850 

51 

30 

30  0 

0 

30  0 

9 

2959 

19 

2957 

28 

2955 

37 

2952 

46 

2949 

55 

31 

31  0 

0 

31  0 

10 

3059 

20 

3057 

30 

3055 

40 

3052 

50 

3049 

59 

32 

32  0 

0 

32  0 

11 

3159 

21 

31  57 

32 

31  55 

43 

31  52 

53 

3148 

7  4 

33 

33  0 

0 

33  0 

12 

3259 

23 

3257 

34 

3255 

46 

3252 

57 

3248 

9 

34 

34  0 

0 

34  0 

12 

3359 

25 

3357 

37 

3354 

49 

3351 

6  1 

3347 

14 

35 

35  0 

0 

35  0 

13 

3459 

26 

3457 

40 

3454 

53 

3451 

6 

3447 

19 

36 

36  0 

0 

36  0 

14 

3558 

28 

3557 

42 

3554 

56 

3551 

10 

3546 

24 

37 

37  0 

0 

37  0 

15 

3658 

30 

3656 

45 

3654 

5  0 

3650 

15 

3646 

30 

38 

38  0 

0 

38  0 

16 

3758 

32 

3756 

48 

3753 

4 

3750 

20 

3745 

36 

39 

39  0 

0 

39  0 

17 

3858 

34 

3856 

51 

3853 

8 

3849 

25 

3845 

42 

40 

40  0 

0 

40  0 

18 

3958 

37 

3956 

55 

3953 

13 

3949 

31 

3944 

49 

41 

41  0 

0 

41  0 

19 

4058 

39 

4056 

58 

4053 

18 

4049 

37 

4044 

56 

42 

42  0 

0 

42  0 

21 

4158 

41 

41  56 

4  2 

41  52 

23 

4148 

43 

4143 

8  3 

43 

43  0 

0 

43  0 

22 

4258 

44 

4256 

6 

4252 

28 

4248 

49 

4242 

11 

44 

44  0 

0 

59 

23 

4358 

47 

4355 

10 

4352 

33 

4347 

56 

4342 

19 

45 

45  0 

0 

4459 

25 

4458 

50 

4455 

14 

4452 

39 

4447 

7  3 

4441 

27 

Table  13.    Kelvin's  Sumner  Line  Table 


293 


b 

a  =  0° 

a  =  1° 

a  =  2° 

a  =  3° 

a  =  4° 

a  =  5° 

a  =  6° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

45  0 

0  0 

4459 

125 

4458 

250 

4455 

414 

4452 

539 

4447 

7'  3 

4441 

827 

46 

46  0 

0 

4559 

26 

4558 

53 

4555 

19 

4551 

45 

4546 

11 

4541 

36 

47 

47  0 

0 

4659 

28 

4658 

56 

4655 

24 

4651 

51 

4646 

19 

4640 

46 

48 

48  0 

0 

4759 

30 

4758 

59 

4755 

29 

4751 

58 

4745 

27 

4739 

56 

49 

49  0 

0 

4859 

31 

4858 

3  3 

4855 

34 

4850 

6  5 

4845 

36 

4838 

9.  6 

50 

50  0 

0 

4959 

33 

4958 

7 

4954 

40 

4950 

13 

4944 

45 

4938 

17 

51 

51  0 

0 

5059 

35 

5057 

11 

5054 

46 

5050 

21 

5044 

55 

5037 

29 

52 

52  0 

0 

5159 

37 

51  57 

15 

51  54 

52 

51  49 

29 

51  43 

8  5 

5136 

41 

53 

53  0 

0 

5259 

40 

5257 

19 

5254 

59 

5249 

38 

5243 

16 

5235 

54 

54 

54  0 

0 

5359 

42 

5357 

24 

5354 

5  6 

5349 

47 

5342 

28 

5334 

10  8 

55 

55  0 

0 

5459 

45 

5457 

29 

5453 

13 

5448 

57 

5441 

40 

5433 

23 

56 

56  0 

0 

5559 

47 

5557 

34 

5553 

21 

5548 

7  8 

5541 

53 

5532 

39 

57 

57  0 

0 

5659 

50 

5657 

40 

5653 

30 

5647 

19 

5640 

9  7 

5631 

55 

58 

58  0 

0 

5759 

53 

5757 

46 

5752 

39 

5747 

31 

5739 

22 

5730 

11  13 

59 

59  0 

0 

5859 

56 

5857 

53 

5852 

49 

5846 

44 

5838 

38 

5829 

32 

60 

60  0 

0 

5959 

2  0 

5956 

4  0 

5952 

59 

5946 

58 

5937 

56 

5928 

52 

61 

61  0 

0 

6059 

4 

6056 

7 

6052 

610 

6045 

813 

6036 

10  14 

6026 

12  14 

62 

62  0 

0 

61  59 

8 

6156 

15 

61  51 

22 

6144 

28 

6135 

33 

61  25 

37 

63 

63  0 

0 

6259 

12 

6256 

24 

6251 

35 

6244 

45 

6234 

54 

6223 

13  2 

64 

64  0 

0 

6359 

17 

6356 

33 

6350 

49 

6343 

9  4 

6333 

11  17 

6322 

29 

65 

65  0 

0 

6459 

22 

6456 

43 

6450 

7  4 

6442 

24 

6432 

42 

6420 

58 

66 

66  0 

0 

6559 

27 

6555 

54 

6549 

20 

6541 

45 

6531 

12  8 

65  18 

1429 

67 

67  0 

0 

6659 

33 

6655 

5  6 

6649 

38 

6640 

10  9 

6629 

37 

66  16 

15  3 

68 

68  0 

0 

6759 

40 

6755 

19 

6748 

58 

6739 

34 

6728 

13  9 

67  14 

40 

69 

69  0 

0 

6859 

47 

6855 

34 

6848 

8  19 

6838 

11  2 

6826 

43 

6812 

1621 

70 

70  0 

0 

6959 

55 

6954 

50 

6947 

43 

6937 

33 

6925 

1421 

69  9 

17  5 

71 

71  0 

0 

7058 

3  4 

7054 

6  7 

7046 

9  9 

7036 

12  7 

7023 

15  2 

70  6 

54 

72 

72  0 

0 

7158 

14 

71  54 

27 

7146 

38 

7135 

45 

71  20 

48 

71  3 

1847 

73 

73  0 

0 

7258 

25 

7253 

49 

7245 

10  10 

7233 

1327 

72  18 

1640 

72  0 

1946 

74 

74  0 

0 

7358 

37 

7353 

7  13 

7344 

46 

7331 

14  14 

73  15 

1737 

56 

2052 

75 

75  0 

0 

7458 

51 

7452 

41 

7443 

11  27 

7429 

15  7 

7412 

1841 

7352 

22  6 

76 

76  0 

0 

7558 

4  8 

7552 

8  13 

7541 

12  13 

7527 

16  7 

75  9 

1953 

7447 

2329 

77 

77  0 

0 

7658 

26 

7651 

50 

7640 

13  7 

7625 

17  16 

76  5 

21  15 

7542 

25  3 

78 

78  0 

0 

7758 

48 

7750 

932 

7738 

14  9 

7722 

1835 

77  1 

2249 

7636 

2649 

79 

79  0 

0 

7857 

514 

7849 

1022 

7836 

1522 

7818 

20  8 

56 

2438 

7729 

2851 

80 

80  0 

0 

7957 

44 

7948 

1122 

7934 

1648 

7914 

21  56 

7850 

2644 

7821 

31  11 

81 

81  0 

0 

8057 

622 

8047 

1235 

8031 

1831 

80  9 

24  5 

7943 

29  13 

7912 

3354 

82 

82  0 

0 

8156 

7  9 

81  45 

14  5 

8128 

2038 

81  4 

2641 

8034 

32  9 

80  1 

37  4 

83 

83  0 

0 

8256 

8  9 

8243 

1559 

8223 

23  16 

57 

2951 

81  24 

3540 

47 

4047 

84 

84  0 

0 

8355 

929 

8341 

1828 

8318 

2638 

8248 

3347 

82  12 

3956 

81  31 

45  9 

85 

85  0 

0 

8454 

11  20 

8437 

2150 

8410 

31  1 

8336 

3844 

56 

45  7 

82  12 

5020 

86 

86  0 

0 

8553 

14  3 

8532 

2636 

85  0 

3655 

8421 

45  4 

8336 

51  26 

48 

5626 

87 

87  0 

0 

8650 

1827 

8624 

3343 

45 

45  2 

85  0 

53  11 

84  10 

59  7 

83  18 

6332 

88 

88  0 

0 

8746 

2634 

87  10 

45  1 

8624 

5620 

32 

6329 

37 

6815 

41 

71  38 

89 

89  0 

0 

8835 

45  0 

46 

6327 

50 

71  35 

53 

7559 

54 

7843 

55 

8034 

90 

90  0 

0 

89  0 

90  0 

88  0 

90  0 

87  0 

90  0 

86  0 

90  0 

85  0 

90  0 

84  0 

90  0 

294 


Table  13.    Kelvin's  Simmer  Line  Table 


b 

a  =  7° 

a  =  8° 

a  =  9° 

a  =  10° 

a  =  11° 

a  =  12° 

a  =  13° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

7  0 

0  0 

8  0 

0  0 

9  0 

0  0 

10  0 

0  0 

11  0 

0  0 

12  0 

0  0 

13  0 

1 

1  0 

0 

59 

0 

59 

0 

59 

0 

59 

0 

59 

0 

58 

0 

2 

59 

0 

159 

0 

159 

0 

158 

0 

158 

0 

157 

0 

1  57 

0 

3 

259 

1 

258 

1 

258 

1 

257 

1 

257 

1 

256 

1 

255 

1 

4 

358 

1 

358 

1 

357 

1 

356 

1 

356 

2 

355 

2 

354 

2 

5 

458 

2 

457 

2 

456 

2 

455 

2 

454 

3 

453 

3 

452 

3 

6 

557 

2 

556 

3 

556 

3 

555 

3 

553 

4 

552 

4 

551 

4 

7 

657 

3 

656 

4 

655 

4 

654 

4 

652 

5 

651 

5 

649 

6 

8 

756 

4 

755 

5 

754 

5 

753 

6 

751 

6 

749 

7 

748 

8 

9 

856 

5 

855 

6 

853 

7 

852 

7 

850 

8 

848 

9 

846 

10 

10 

955 

6 

954 

7 

953 

8 

951 

9 

949 

10 

947 

11 

944 

12 

11 

1055 

8 

1053 

9 

1052 

10 

1050 

11 

1048 

12 

1045 

13 

1043 

14 

12 

11  55 

9 

1153 

11 

11  51 

12 

1149 

13 

1147 

14 

1144 

16 

11  41 

17 

13 

1254 

11 

1252 

13 

1250 

14 

1248 

15 

1245 

17 

1243 

19 

1240 

20 

14 

1354 

13 

1352 

15 

1349 

16 

1347 

18 

1344 

20 

1341 

22 

1338 

23 

15 

1453 

15 

1451 

17 

1449 

19 

1446 

21 

1443 

23 

1440 

25 

1436 

26 

16 

1553 

17 

1550 

19 

1548 

21 

1545 

24 

1542 

26 

1538 

28 

1535 

30 

17 

1652 

19 

1650 

22 

1647 

24 

1644 

27 

1641 

29 

1637 

32 

1633 

34 

18 

1752 

21 

1749 

24 

1746 

27 

1743 

30 

1739 

33 

1736 

36 

1731 

39 

19 

1851 

24 

1848 

27 

1845 

30 

1842 

34 

1838 

37 

1834 

40 

1830 

43 

20 

1951 

27 

1948 

30 

1945 

34 

1941 

38 

1937 

41 

1933 

45 

1928 

48 

21 

2050 

30 

2047 

34 

2044 

38 

2040 

42 

2036 

46 

2031 

50 

2026 

53 

22 

2150 

33 

2146 

37 

2143 

42 

2139 

46 

21  35 

51 

2130 

55 

2124 

59 

23 

2249 

36 

2246 

41 

2242 

46 

2238 

51 

2233 

56 

2228 

13  0 

2223 

14  5 

24 

2349 

39 

2345 

45 

2341 

50 

2337 

56 

2332 

12  1 

2327 

6 

2321 

11 

25 

2448 

43 

2444 

49 

2440 

55 

2436 

11  1 

2431 

6 

2425 

12 

2419 

17 

26 

2548 

47 

2544 

53 

2539 

10  0 

2535 

6 

2529 

12 

2523 

18 

2517 

24 

27 

2647 

51 

2643 

58 

2638 

5 

2633 

12 

2628 

18 

2622 

25 

2615 

31 

28 

2746 

55 

2742 

9  3 

2738 

10 

2732 

18 

2727 

25 

2720 

32 

27  13 

39 

29 

2846 

59 

2841 

8 

2837 

16 

2831 

24 

2825 

32 

2818 

40 

28  11 

47 

30 

2945 

8  4 

2941 

13 

2936 

22 

2930 

31 

2924 

39 

2917 

48 

29  9 

56 

31 

3045 

9 

3040 

19 

3035 

28 

3029 

38 

3022 

47 

3015 

56 

30  7 

15  5 

32 

3144 

14 

31  39 

25 

3134 

35 

3127 

45 

3121 

55 

31  13 

14  4 

31  5 

14 

33 

3243 

20 

3238 

31 

3233 

42 

3226 

52 

3219 

13  3 

3211 

13 

32  3 

24 

34 

3343 

26 

3337 

37 

3332 

49 

3325 

12  0 

3318 

12 

3310 

23 

33  1 

34 

35 

3442 

32 

3437 

44 

3430 

57 

3424 

9 

3416 

21 

34  8 

33 

59 

44 

36 

3541 

38 

3536 

51 

3529 

11  5 

3522 

18 

3514 

31 

35  6 

43 

3456 

55 

37 

3641 

45 

3635 

59 

3628 

13 

3621 

27 

3613 

41 

36  4 

54 

3554 

16  7 

38 

3740 

52 

3734 

10  7 

3727 

22 

3719 

37 

3711 

51 

37  2 

15  6 

3652 

20 

39 

3839 

59 

3833 

15 

3826 

31 

3818 

47 

38  9 

14  2 

38  0 

18 

3749 

33 

40 

3939 

9  6 

3932 

24 

3925 

41 

3916 

58 

39  7 

14 

57 

31 

3847 

46 

41 

4038 

14 

4031 

33 

4023 

51 

4015 

13  9 

40  5 

27 

3955 

44 

3944 

17  0 

42 

4137 

23 

4130 

43 

4122 

12  2 

41  13 

21 

41  3 

40 

4053 

58 

4041 

15 

43 

4236 

32 

4229 

53 

4221 

13 

4212 

33 

42  1 

53 

41  51 

1612 

41  39 

31 

44 

4335 

41 

4328 

11  3 

43  19 

25 

4310 

46 

59 

15  7 

4248 

28 

4236 

48 

45 

4434 

51 

4427 

14 

4418 

38 

44  8 

14  0 

4357 

22 

4346 

44 

4333 

18  5 

Table  13.    Kelvin's  Sumner  Line  Table 


295 


b 

a  =  7° 

a=8° 

a  =  9° 

a  =  10° 

a  =  11  ° 

a  =  12° 

a  =  13° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

4434 

951 

4427 

11  14 

4418 

1238 

44  8 

14  0 

4357 

1522 

4346 

1644 

4333 

18  5 

46 

4534 

10  1 

4526 

26 

4516 

51 

45  6 

15 

4455 

38 

4443 

17  1 

4430 

23 

47 

4633 

12 

4624 

39 

46  15 

13  5 

46  4 

30 

4553 

55 

4540 

19 

4527 

42 

48 

4732 

24 

4723 

52 

47  13 

19 

47  2 

46 

4651 

16  12 

4638 

37 

4624 

19  2 

49 

4831 

36 

4822 

12  6 

48  12 

34 

48  0 

15  3 

4748 

30 

4735 

57 

4720 

23 

50 

4930 

49 

4920 

20 

4910 

51 

58 

20 

4846 

50 

4832 

1818 

48  17 

45 

51 

5029 

11  2 

50  19 

35 

50  8 

14  8 

4956 

39 

4943 

17  10 

4929 

40 

49  13 

20  9 

52 

51  27 

17 

51  18 

52 

51  6 

26 

5054 

59 

5040 

31 

5026 

19  3 

50  9 

33 

53 

5226 

32 

52  16 

13  9 

52  4 

45 

51  52 

1620 

51  37 

54 

51  22 

27 

51  5 

59 

54 

5325 

48 

5314 

27 

53  2 

15  5 

5249 

42 

5234 

1818 

52  19 

53 

52  1 

21  27 

55 

5424 

12  5 

54  13 

46 

54  0 

26 

5347 

17  5 

5331 

43 

53  15 

2020 

57 

56 

56 

5522 

23 

55  11 

14  6 

58 

49 

5444 

30 

5428 

19  10 

54  11 

49 

5353 

2226 

57 

5621 

42 

56  9 

28 

5556 

1613 

5541 

56 

5525 

39 

55  7 

21  19 

5448 

58 

58 

5719 

13  3 

57  7 

51 

5653 

39 

5638 

1824 

5621 

20  9 

56  3 

51 

5543 

2333 

59 

5818 

25 

58  5 

1516 

5751 

17  6 

5735 

54 

5717 

41 

59 

2226 

5638 

24  9 

60 

59  16 

48 

59  3 

42 

5848 

35 

5832 

1926 

58  13 

21  15 

5754 

23  2 

5733 

47 

61 

6014 

1413 

60  1 

1610 

5945 

18  6 

5928 

59 

59  9 

51 

5849 

41 

5827 

2528 

62 

61  12 

39 

58 

40 

6042 

39 

6024 

2035 

60  5 

2230 

5944 

2422 

5921 

26  11 

63 

62  10 

15  8 

6156 

17  12 

61  39 

19  14 

6120 

21  14 

61  0 

23  11 

6038 

25  5 

6015 

57 

64 

63  8 

39 

6253 

47 

6236 

52 

62  16 

55 

55 

55 

6132 

52 

61  8 

2746 

65 

64  6 

1612 

6350 

1824 

6332 

2033 

63  12 

2239 

6250 

2442 

6226 

2642 

62  1 

2839 

66 

65  4 

48 

6447 

19  4 

6428 

21  17 

64  7 

2326 

6344 

2533 

6320 

2736 

53 

2935 

67 

66  1 

1727 

6543 

47 

6524 

22  4 

65  2 

2417 

6438 

2627 

6413 

2833 

6345 

3035 

68 

58 

18  9 

6639 

2034 

66  19 

55 

56 

25  12 

6532 

2726 

65  5 

29  34 

6437 

31  39 

69 

6755 

55 

6735 

21  25 

6714 

2351 

6650 

26  12 

6625 

2829 

57 

3040 

6528 

3247 

70 

6852 

1945 

6831 

2220 

68  9 

2451 

6744 

2716 

6717 

2937 

6648 

31  52 

6618 

34  1 

71 

6948 

2040 

6926 

2321 

69  3 

2556 

6837 

2826 

68  9 

3050 

6739 

33  8 

67  7 

3520 

72 

7044 

21  40 

7021 

2427 

57 

27  8 

6929 

2943 

69  0 

32  10 

6829 

3431 

55 

3646 

73 

7139 

2247 

71  16 

2540 

7050 

2827 

7021 

31  6 

50 

33  37 

69  18 

36  1 

6843 

38  18 

74 

7234 

24  1 

72  10 

27  1 

7142 

2953 

71  12 

3236 

7040 

35  12 

70  6 

3738 

6930 

3957 

75 

7329 

2523 

73  3 

2830 

7234 

31  28 

72  2 

3416 

7128 

3655 

53 

3924 

7015 

41  44 

76 

7423 

2655 

55 

30  9 

7324 

33  13 

51 

36  5 

72  16 

3847 

7138 

41  18 

59 

4340 

77 

7516 

2838 

7446 

32  0 

74  14 

35  9 

7339 

38  5 

73  2 

4050 

7223 

4323 

71  42 

4545 

78 

76  8 

3034 

7537 

34  4 

75  2 

3718 

7426 

40  18 

47 

43  4 

73  6 

4538 

7223 

48  0 

79 

59 

3246 

7626 

3623 

49 

3942 

75  11 

4244 

7430 

4532 

47 

48  5 

73  2 

5026 

80 

7749 

3516 

7713 

3859 

7635 

4222 

54 

4526 

7511 

4813 

7426 

5045 

39 

53  3 

81 

7837 

38  8 

59 

41  56 

77  18 

4521 

7635 

4825 

49 

51  10 

75  2 

5339 

74  14 

5553 

82 

7923 

41  25 

7842 

45  17 

59 

4842 

77  13 

5143 

7626 

5424 

37 

5647 

46 

5855 

83 

80  7 

45  13 

7923 

49  4 

7837 

5225 

49 

5521 

59 

5755 

76  8 

60  10 

75  16 

62  10 

84 

47 

4936 

80  1 

5322 

7912 

5635 

7821 

5920 

7729 

6144 

36 

6349 

42 

6538 

85 

8124 

5438 

35 

5812 

43 

61  11 

50 

6342 

56 

6551 

77  1 

6742 

76  5 

6919 

86 

57 

6024 

81  4 

6336 

80  9 

66  14 

79  14 

6825 

7818 

70  16 

22 

71  50 

25 

73  11 

87 

8223 

6655 

28 

6935 

31 

7143 

34 

7328 

36 

7456 

38 

76  10 

40 

77  14 

88 

43 

74  8 

45 

76  3 

47 

7734 

48 

7848 

49 

7949 

50 

8041 

51 

81  24 

89 

56 

81  55 

56 

8255 

57 

8343 

57 

8421 

57 

8452 

58 

8518 

58 

8541 

90 

83  0 

90  0 

82  0 

90  0 

81  0 

90  0 

80  0 

90  0 

79  0 

90  0 

78  0 

90  0 

77  0 

90  0 

296 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  =  14° 

a  =  15° 

a  =  16° 

a  =  17° 

a  =  18° 

a=  19° 

a  =  20° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

1 

0  0 

14  0 

0  0 

15  0 

0  0 

16  0 

0  0 

17  0 

0  0 

18  0 

0  0 

19  0 

0  0 

20  0 

58 

0 

58 

0 

58 

0 

57 

0 

57 

0 

57 

0 

56 

0 

2 

156 

0 

156 

• 

1  55 

155 

; 

1  54 

154 

1  53 

1 

J 

255 

] 

254 

• 

253 

• 

252 

• 

251 

• 

250 

2 

249 

2 

4 

353 

t 

352 

2 

351 

2 

349 

t 

348 

t 

347 

3 

346 

3 

5 

451 

3 

450 

t 

448 

3 

447 

i 

445 

^ 

444 

^ 

442 

4 

( 

549 

4 

548 

c 

t. 

546 

i 

c 

544 

i 

«. 

542 

( 

540 

6 

538 

6 

f 

647 

6 

646 

r 

644 

*" 

642 

*j 

639 

8 

637 

t 

635 

8 

8 

746 

8 

744 

C 

741 

c 

739 

9 

736 

10 

734 

10 

731 

11 

c 

844 

10 

841 

11 

839 

11 

836 

12 

833 

13 

830 

13 

827 

14 

10 

942 

12 

939 

13 

937 

14 

934 

15 

930 

16 

927 

16 

923 

17 

11 

1040 

15 

1037 

16 

1034 

17 

1031 

18 

1027 

19 

1024 

20 

1020 

21 

12 

1138 

18 

1135 

19 

1132 

20 

1128 

21 

11  24 

23 

11  20 

2^ 

11  16 

25 

13 

1236 

21 

1233 

22 

1229 

24 

1225 

25 

1221 

27 

12  17 

28 

1212 

29 

14 

1335 

25 

1331 

26 

1327 

28 

1323 

29 

1318 

31 

13  13 

32 

13  8 

34 

15 

1433 

28 

1429 

30 

1424 

32 

1420 

34 

14  15 

36 

14  10 

37 

14  5 

39 

16 

1531 

32 

1526 

35 

1522 

37 

15  17 

39 

15  12 

41 

15  6 

42 

15  1 

44 

17 

1629 

37 

1624 

39 

16  19 

42 

16  14 

44 

16  9 

46 

16  3 

48 

57 

50 

18 

1727 

41 

1722 

44 

17  17 

47 

1711 

49 

17  6 

52 

59 

54 

1653 

56 

19 

1825 

46 

1820 

49 

18  14 

52 

18  8 

55 

18  2 

58 

1756 

20  1 

1749 

21  3 

20 

1923 

52 

19  17 

55 

19  12 

58 

19  5 

18  1 

59 

19  4 

1852 

8 

1845 

10 

21 

2021 

57 

20  15 

16  1 

20  9 

17  4 

20  2 

8 

1956 

11 

1948 

15 

1941 

•  18 

22 

21  19 

15  3 

21  13 

7 

21  6 

11 

59 

15 

2052 

19 

2045 

22 

2037 

26 

23 

22  17 

g 

2210 

14 

22  4 

18 

21  56 

22 

21  49 

27 

21  41 

30 

21  32 

34 

24 

2315 

16 

23  8 

21 

23  1 

26 

2253 

30 

2245 

35 

2237 

39 

2228 

43 

25 

24  13 

23 

24  6 

28 

58 

34 

2350 

38 

2342 

43 

2333 

48 

2324 

53 

26 

25  10 

30 

25  3 

36 

2455 

42 

2447 

47 

2438 

52 

2429 

58 

2420 

22  3 

27 

26  8 

38 

26  1 

44 

2552 

50 

2544 

56 

2535 

20  2 

2525 

21  8 

25  15 

13 

28 

27  6 

46 

58 

53 

2650 

59 

2641 

19  6 

2631 

12 

2621 

18 

2611 

24 

29 

28  4 

55 

2755 

17  2 

2747 

18  9 

2737 

16 

2727 

23 

2717 

29 

27  6 

36 

30 

29  1 

16  4 

853 

12 

2844 

19 

2834 

27 

2824 

34 

28  13 

41 

28  2 

48 

31 

59 

13 

950 

22 

2941 

30 

2930 

38 

2920 

46 

29  9 

53 

57 

23  1 

32 

3057 

23 

047 

32 

3037 

41 

3027 

50 

3016 

58 

30  4 

22  6 

2952 

14 

33 

31  54 

33 

1  44 

43 

1  34 

53 

3123 

20  2 

31  12 

21  11 

31  0 

19 

3047 

28 

34 

3252 

44 

242 

55 

231 

19  5 

3220 

15 

32  8 

24 

55 

33 

3142 

42 

35 

3349 

56 

339 

8  7 

328 

18 

3  16 

28 

3  4 

38 

3251 

48 

3237 

57 

36 

3446 

7  8 

436 

20 

424 

31 

4  12 

42 

59 

53 

3346 

23  3 

3332 

24  13 

37 

544 

20 

533 

33 

521 

45 

5  8 

57 

455 

22  8 

441 

19 

3426 

30 

38 

641 

33 

629 

47 

617 

20  0 

6  4 

21  12 

550 

24 

536 

36 

521 

48 

39 

738 

47 

726 

9  1 

713 

15 

7  0 

28 

646 

41 

631 

54 

6  15 

25  6 

40 

835 

8  2 

823 

17 

8  10 

31 

56 

45 

741 

59 

726 

24  12 

7  9 

25 

41 

932 

17 

9  19 

33 

9  6 

48 

852 

22  3 

836 

23  18 

820 

32 

8  3 

45 

42 

029 

33 

0  16 

50 

0  2 

21  6 

947 

22 

931 

37 

9  15 

52 

57 

26  6 

43 

1  26 

50 

1  12 

20  7 

58 

25 

042 

41 

026 

57 

0  9 

25  13 

951 

27 

44 

223 

9  7 

2  9 

26 

154 

44 

138 

23  2 

1  21 

24  18 

1  3 

35 

045 

50 

45 

3  19 

25 

3  5 

45 

249 

22  4 

233 

23 

2  16 

41 

57 

58 

139 

2714 

Table  13.    Kelvin's  Sumner  Line  Table 


297 


b 

a  =  14° 

a  =  15° 

a  =  16° 

a  =  17° 

a  =  18° 

a  =  19° 

a  =  20° 

K 

„ 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

43  19 

1925 

43  5 

2045 

4249 

22  4 

4233 

2323 

42  16 

2441 

41  57 

2558 

41  39 

27  14 

46 

44  16 

45 

44  1 

21  6 

4345 

26 

4328 

45 

43  10 

25  4 

4251 

2G12L 

4232 

39 

47 

45  12 

20  5 

57 

27 

4440 

48 

4423 

24  9 

44  4 

28 

4345 

47 

4325 

28  5 

48 

46  8 

26 

4553 

49 

4535 

23  12 

4518 

33 

58 

54 

44  39 

27  14 

44  18 

33 

49 

47  4 

48 

4648 

2213 

4630 

37 

4612 

59 

4552 

2621 

4532 

42 

4510 

29  1 

50 

48  0 

21  12 

4744 

38 

4725 

24  3 

47  6 

2526 

4646 

49 

4625 

28  11 

46  2 

31 

5] 

56 

37 

4839 

23  4 

4820 

30 

48  0 

55 

4739 

2718 

47  18 

41 

54 

30  3 

5S 

4952 

22  3 

4934 

31 

4915 

59 

54 

2625 

4832 

49 

48  10 

29  13 

4746 

36 

53 

5048 

30 

5029 

24  0 

50  9 

2529 

4948 

56 

4925 

2822 

49  2 

47 

4838 

31  10 

54 

5143 

59 

51  24 

30 

51  3 

26  0 

5041 

2729 

50  18 

56 

54 

3022 

4929 

46 

55 

5238 

2330 

5218 

25  2 

57 

34 

51  34 

28  4 

51  10 

2932 

5046 

59 

5020 

3224 

56 

5333 

24  2 

5312 

36 

5250 

27  9 

5227 

40 

52  2 

3010 

51  37 

31  37 

51  10 

33  3 

57 

5428 

36 

54  6 

2612 

5343 

46 

5320 

29  18 

54 

49 

5228 

32  18 

52  0 

45 

58 

5522 

25  12 

55  0 

49 

5436 

2825 

54  12 

59 

5346 

31  31 

53  18 

33  1 

50 

3429 

59 

56  16 

50 

53 

2729 

5529 

29  6 

55  4 

3042 

5437 

32  15 

54  8 

46 

5339 

35  15 

60 

5710 

2630 

5646 

28  11 

5621 

50 

55 

31  27 

5527 

33  1 

58 

3433 

5428 

36  3 

61 

58  4 

2713 

5739 

56 

57  13 

3036 

5646 

32  14 

56  17 

50 

5547 

3523 

55  16 

54 

62 

57 

59 

5831 

2943 

58  4 

3125 

5736 

33  4 

57  7 

3441 

5636 

36  16 

56  4 

3747 

63 

5950 

2847 

5923 

3033 

55 

3217 

5826 

57 

56 

3535 

5724 

37  11 

51 

3843 

64 

6042 

2938 

6015 

3126 

5946 

3311 

59  16 

3453 

5844 

3633 

58  11 

38  9 

5738 

3942 

65 

61  34 

3032 

61  6 

3223 

6036 

34  9 

60  5 

3553 

5932 

3733 

58 

39  10 

5824 

4044 

66 

6225 

31  31 

56 

3323 

6125 

3511 

53 

3656 

6019 

3837 

5944 

40  15 

59  9 

41  49 

67 

63  16 

3233 

6246 

3427 

62  14 

36  17 

61  41 

38  3 

61  6 

3945 

6030 

41  23 

53 

4258 

68 

64  7 

33  39 

6335 

3535 

63  2 

3726 

6228 

39  13 

52 

4056 

61  15 

4235 

6036 

4411 

69 

56 

3450 

6423 

3647 

49 

3840 

63  14 

4028 

6237 

42  12 

58 

4351 

61  19 

4527 

70 

6545 

36  5 

6511 

38  4 

6436 

3958 

59 

4148 

6321 

4332 

6241 

4511 

62  1 

4647 

71 

6633 

3727 

58 

3927 

6521 

41  22 

6443 

43  12 

64  4 

4456 

6323 

4636 

41 

4811 

72 

6720 

3854 

6644 

4056 

66  6 

4252 

6526 

4442 

45 

4626 

64  4 

48  6 

6321 

4940 

73 

68  7 

4027 

6729 

4230 

49 

4427 

66  8 

46  17 

6526 

48  1 

43 

4940 

59 

51  14 

74 

52 

42  8 

68  12 

44  11 

6731 

46  8 

49 

4758 

66  6 

4942 

6521 

51  19 

6436 

5252 

75 

6936 

4356 

55 

4559 

6812 

4756 

6729 

4945 

44 

5128 

58 

53  4 

65  11 

5435 

76 

7018 

4552 

6936 

4755 

52 

4951 

68  7 

51  39 

6720 

5320 

6633 

5455 

45 

5623 

77 

59 

4757 

7015 

4959 

6930 

51  53 

43 

5339 

55 

55  18 

67  7 

5651 

66  18 

5817 

78 

7138 

50  11 

53 

52  12 

70  6 

54  3 

69  18 

5547 

6829 

5723 

39 

5853 

48 

60  16 

79 

7216 

5235 

71  28 

5433 

40 

5621 

50 

58  2 

69  0 

5935 

68  9 

61  0 

67  17 

6220 

80 

51 

55  9 

72  2 

57  3 

71  12 

5848 

7021 

6024 

29 

61  53 

37 

63  14 

44 

6430 

81 

7324 

5754 

34 

5943 

42 

61  23 

50 

6254 

56 

64  18 

69  3 

6534 

68  9 

6645 

82 

55 

6050 

73  3 

6233 

72  9 

64  7 

71  16 

6531 

7021 

6649 

27 

68  0 

31 

69  5 

83 

7423 

6357 

29 

6532 

34 

6659 

39 

68  16 

44 

6926 

48 

7031 

51 

71  29 

84 

48 

67  15 

52 

6841 

56 

6959 

72  0 

71  7 

71  3 

7210 

70  7 

73  7 

69  9 

7359 

85 

75  9 

7044 

7412 

71  59 

73  15 

73  6 

18 

74  5 

20 

7459 

23 

7548 

25 

7632 

86 

27 

7422 

29 

7525 

31 

7620 

33 

H  9 

35 

7753 

36 

7833 

37 

79  9 

87 

41 

78  9 

43 

7857 

44 

7939 

45 

80  17 

46 

8051 

46 

81  22 

47 

81  49 

88 

52 

82  2 

52 

8235 

53 

83  4 

53 

8329 

54 

8352 

54 

8413 

54 

8431 

89 

58 

86  0 

58 

8616 

58 

8631 

58 

8644 

58 

8655 

58 

87  6 

59 

87  15 

90 

76  0 

90  0 

75  0 

90  0 

74  0 

90  0 

73  0 

90  0 

72  0 

90  0 

71  0 

90  0 

70  0 

90  0 

298 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  =  21° 

a  =  22° 

a  =  23° 

a  =  24° 

a  =  25° 

a  =  26° 

a  =  27° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

21  0 

0  0 

22  0 

0  0 

23  0 

0  0 

24  0 

0  0 

25  0 

0  0 

26  0 

0  0 

27  0 

1 

56 

0 

56 

0 

55 

0 

55 

0 

54 

0 

54 

0 

53 

0 

2 

152 

1 

1  51 

1 

1  51 

1 

1  50 

1 

1  49 

1 

148 

1 

1  47 

1 

3 

248 

2 

247 

2 

246 

2 

244 

2 

243 

2 

242 

2 

240 

2 

4 

344 

3 

342 

3 

341 

3 

339 

3 

338 

3 

336 

3 

334 

3 

5 

440 

4 

438 

5 

436 

5 

434 

5 

432 

5 

430 

5 

427 

5 

6 

536 

6 

534 

7 

531 

7 

529 

7 

526 

7 

523 

7 

521 

8 

7 

632 

9 

629 

9 

626 

9 

624 

10 

620 

10 

617 

10 

614 

11 

8 

728 

11 

725 

12 

722 

12 

7  18 

13 

715 

13 

711 

13 

7  7 

14 

9 

824 

14 

820 

15 

817 

15 

813 

16 

8  9 

16 

8  5 

17 

8  1 

17 

10 

920 

18 

916 

18 

912 

19 

9  8 

20 

9  3 

20 

59 

21 

54 

21 

11 

1016 

22 

1011 

22 

10  7 

23 

10  2 

24 

57 

24 

953 

25 

947 

26 

12 

11  12 

26 

11  7 

26 

11  2 

27 

57 

28 

1052 

29 

1046 

30 

1041 

31 

13 

12  7 

30 

12  2 

31 

57 

32 

11  52 

33 

11  46 

34 

11  40 

35 

11  34 

36 

14 

13  3 

35 

58 

36 

1252 

38 

1246 

39 

1240 

40 

1234 

41 

1227 

42 

15 

59 

40 

1353 

42 

1347 

43 

1341 

45 

1334 

46 

1327 

47 

1320 

49 

16 

1455 

46 

1448 

48 

1442 

49 

1435 

51 

1428 

53 

1421 

54 

14  13 

56 

17 

1550 

52 

1544 

54 

1537 

56 

1529 

58 

1522 

26  0 

1514 

27  1 

15  6 

28  3 

18 

1646 

59 

1639 

23  1 

1632 

24  3 

1624 

25  5 

1616 

7 

16  8 

9 

59 

11 

19 

1742 

22  6 

1734 

8 

1726 

11 

17  18 

13 

1710 

15 

17  1 

17 

1652 

19 

20 

1837 

13 

1829 

16 

1821 

19 

1812 

21 

18  3 

23 

54 

26 

1745 

28 

21 

1933 

21 

1924 

24 

19  16 

27 

19  7 

30 

57 

32 

1847 

35 

1837 

37 

22 

2028 

29 

2019 

33 

20  10 

36 

20  1 

39 

1951 

42 

1940 

45 

1930 

47 

23 

2124 

38 

21  14 

42 

21  5 

45 

55 

49 

2044 

52 

2033 

55 

2022 

58 

24 

22  19 

47 

22  9 

52 

59 

55 

21  49 

59 

21  38 

27  3 

21  26 

28  6 

21  15 

29  9 

25 

23  14 

57 

23  4 

24  2 

2254 

25  6 

2243 

26  10 

2231 

14 

22  19 

17 

22  7 

21 

26 

24  9 

23  8 

59 

12 

2348 

17 

2337 

21 

2325 

25 

2312 

29 

23  0 

33 

27 

25  4 

19 

2454 

23 

2442 

28 

2430 

33 

2418 

37 

24  5 

42 

52 

46 

28 

59 

30 

2548 

35 

2536 

40 

2524 

46 

2511 

50 

57 

55 

2444 

59 

29 

2654 

42 

2643 

48 

2630 

53 

26  17 

59 

26  4 

28  4 

2550 

29  9 

2536 

30  13 

30 

2749 

54 

2737 

25  1 

2724 

26  7 

2711 

2713 

57 

18 

2642 

23 

2627 

28 

31 

2844 

24  7 

2832 

14 

28  18 

21 

28  4 

27 

2750 

33 

2734 

38 

2719 

44 

32 

2939 

21 

2926 

28 

29  12 

35 

57 

42 

2842 

48 

2826 

54 

2811 

31  0 

33 

3034 

36 

3020 

43 

30  5 

51 

2950 

58 

2935 

29  4 

2918 

3011 

29  2 

17 

34 

31  28 

51 

31  14 

59 

59 

27  7 

3043 

28  14 

3027 

21 

3010 

28 

53 

35 

35 

3223 

25  7 

32  8 

2615 

3152 

24 

3136 

31 

31  19 

39 

31  2 

46 

3044 

53 

36 

33  17 

23 

33  1 

32 

3245 

41 

3229 

49 

3211 

58 

53 

31  5 

3135 

32  12 

37 

3411 

40 

55 

50 

3338 

59 

3321 

29  8 

33  3 

3017 

3245 

25 

3226 

32 

38 

35  5 

58 

3448 

27  9 

3431 

28  18 

3413 

28 

55 

37 

3336 

45 

3316 

53 

39 

59 

26  17 

3542 

28 

3524 

38 

35  5 

49 

3447 

58 

3427 

32  7 

34  6 

3315 

40 

3653 

37 

3635 

48 

3617 

59 

57 

3010 

3538 

3120 

3518 

29 

56 

38 

41 

3746 

58 

3728 

28  10 

37  9 

2921 

3649 

32 

3629 

43 

36  8 

52 

3546 

34  1 

42 

3840 

27  19 

3821 

32 

38  1 

44 

3741 

56 

3720 

32  7 

58 

3317 

3636 

26 

43 

3933 

42 

39  13 

55 

53 

30  8 

3832 

31  20 

3811 

31 

3748 

42 

3725 

52 

44 

4026 

28  5 

40  6 

29  19 

3945 

33 

3923 

45 

39  1 

57 

3838 

34  8 

3814 

35  19 

45 

41  19 

30 

58 

44 

4037 

59 

4014 

3212 

51 

3324 

3928 

36 

39  3 

47 

Table  13.    Kelvin's  Sunnier  Line  Table 


299 


b 

a  =  31° 

a  =  23° 

a  =  23° 

a  =  24° 

a  =  25° 

a  =  26° 

a  =  27° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

41  19 

2830 

4058 

2944 

4037 

3059 

40  14 

32  12 

3951 

3324 

3928 

3436 

39  3 

3547 

46 

4211 

55 

4150 

3011 

41  28 

31  26 

41  5 

39 

4041 

52 

40  17 

35  4 

52 

36  16 

47 

43  4 

2922 

4242 

39 

4219 

54 

55 

33  8 

4131 

3422 

41  6 

34 

4040 

46 

48 

56 

50 

4333 

31  8 

4310 

3223 

4245 

38 

4220 

52 

55 

36  5 

41  28 

3717 

49 

4448 

3020 

4424 

38 

44  0 

54 

4335 

3410 

43  9 

3524 

4243 

37 

42  15 

50 

50 

4540 

51 

4515 

32  9 

50 

3326 

4425 

43 

58 

57 

4331 

3711 

43  2 

3824 

51 

4631 

31  23 

46  6 

42 

4540 

34  0 

45  14 

3517 

4447 

3632 

4418 

46 

49 

39  0 

52 

4722 

57 

56 

33  17 

4630 

35 

46  3 

53 

4535 

37  8 

45  5 

3823 

4436 

37 

53 

48  13 

3232 

4746 

53 

47  19 

35  12 

51 

3630 

4622 

46 

52 

39  1 

4522 

4015 

54 

49  3 

33  9 

4836 

3430 

48  8 

50 

4739 

37  9 

47  9 

3826 

4639 

41 

46  7 

55 

55 

53 

48 

4925 

35  10 

56 

3630 

4827 

49 

56 

39  7 

4725 

4023 

52 

4137 

56 

5043 

3428 

5014 

51 

4944 

37  12 

49  14 

3832 

4842 

49 

48  10 

41  6 

4737 

4220 

57 

5132 

35  11 

51  2 

3634 

5032 

56 

50  1 

39  16 

4928 

4034 

55 

51 

4821 

43  5 

58 

5221 

55 

50 

3719 

51  19 

3842 

47 

40  2 

5014 

41  21 

4940 

4238 

49  5 

52 

59 

53  9 

3642 

5238 

38  7 

52  6 

3930 

5133 

50 

59 

42  9 

5024 

4326 

48 

4441 

60 

57 

3731 

5325 

56 

52 

4020 

52  18 

4141 

5143 

43  0 

51  7 

44  17 

5030 

4532 

61 

5444 

3822 

5411 

3948 

5337 

41  12 

53  2 

4234 

5226 

53 

49 

45  10 

51  12 

4625 

62 

5531 

3916 

57 

4043 

5422 

42  7 

46 

4329 

53  9 

4448 

5231 

46  6 

53 

4721 

63 

5617 

40  13 

5542 

41  40 

55  6 

43  5 

5429 

4427 

51 

4546 

53  13 

47  3 

5233 

4818 

64 

57  3 

41  13 

5627 

4240 

49 

44  5 

5511 

4527 

5433 

4646 

53 

48  3 

53  13 

4918 

65 

48 

4215 

5711 

4343 

5632 

45  8 

53 

4630 

5513 

4749 

5433 

49  5 

51 

5020 

66 

5832 

4321 

54 

4449 

5714 

46  13 

5634 

4735 

53 

4854 

5512 

5010 

5429 

5124 

67 

5915 

4430 

5836 

4558 

55 

4722 

5714 

4844 

5632 

50  2 

50 

51  18 

55  6 

5231 

68 

57 

4542 

59  17 

47  10 

5835 

4834 

53 

4955 

57  10 

51  13 

5627 

52  28 

42 

5341 

69 

6039 

4658 

57 

4826 

5915 

4950 

5831 

51  10 

47 

5227 

57  3 

5342 

56  17 

5453 

70 

61  19 

4818 

6036 

4945 

53 

51  8 

59  8 

5228 

5823 

5344 

38 

5458 

51 

56  8 

71 

58 

4942 

61  14 

51  8 

6030 

5231 

44 

5349 

58 

55  5 

58  12 

56  17 

5724 

5725 

72 

6236 

51  10 

51 

5235 

61  6 

5357 

6019 

55  14 

5932 

5628 

44 

5739 

56 

5846 

73 

63  13 

5242 

6227 

54  7 

41 

5527 

53 

5642 

60  5 

5755 

5916 

59  4 

5826 

60  9 

74 

49 

54  19 

63  2 

5542 

6214 

57  0 

6125 

5814 

36 

5925 

46 

6032 

55 

6135 

75 

6423 

56  1 

35 

5721 

46 

5838 

56 

5950 

61  6 

6058 

60  15 

62  3 

5923 

63  4 

76 

56 

5747 

64  7 

59  5 

6316 

6019 

6226 

61  29 

34 

6235 

42 

6337 

50 

6436 

77 

6527 

5938 

37 

6053 

45 

62  5 

54 

63  12 

62  1 

6415 

61  8 

65  14 

6015 

6611 

78 

57 

61  34 

65  5 

6246 

6413 

6354 

6320 

6458 

26 

6558 

32 

6655 

38 

6748 

79 

6625 

6334 

32 

6443 

38 

6548 

44 

6648 

50 

6745 

55 

6838 

61  0 

6928 

80 

50 

6540 

56 

6644 

65  2 

6745 

64  7 

6842 

63  12 

6935 

62  16 

7024 

20 

71  11 

81 

6714 

6750 

66  19 

6850 

23 

6946 

28 

7039 

32 

71  27 

35 

72  13 

39 

7256 

82 

36 

70  4 

40 

71  0 

43 

7151 

47 

7239 

50 

7323 

53 

74  4 

56 

7443 

83 

55 

7223 

58 

73  13 

66  1 

7359 

65  3 

7442 

64  6 

7521 

63  8 

7558 

62  10 

7633 

84 

6812 

7446 

6714 

7530 

16 

7610 

18 

7647 

20 

7722 

22 

7754 

23 

7824 

85 

26 

7712 

28 

7750 

29 

7824 

31 

7855 

32 

7925 

33 

7952 

35 

8018 

86 

38 

7942 

39 

8012 

40 

8040 

41 

81  6 

42 

8130 

43 

8152 

44 

8212 

87 

48 

82  14 

48 

8237 

49 

8258 

49 

8318 

50 

8336 

50 

8353 

51 

84  8 

88 

55 

8448 

55 

85  4 

55 

8518 

55 

8531 

56 

8543 

56 

8555 

56 

86  5 

89 

59 

8724 

59 

8732 

59 

8739 

59 

8745 

59 

8751 

59 

8757 

59 

88  2 

90 

69  0 

90  0 

68  0 

90  0 

67  0 

90  0 

66  0 

90  0 

65  0 

90  0 

64  0 

90  0 

63  0 

90  0 

300 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  =  28° 

a  =  29° 

a  =  30° 

a  =  313 

a  =  32° 

a  =  33° 

a  =  34° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

28  0 

0  0 

29  0 

0  0 

30  0 

0  0 

31  0 

0  0 

32  0 

0  0 

33  0 

0  0 

34  0 

1 

53 

0 

52 

0 

52 

0 

51 

0 

51 

0 

50 

0 

50 

0 

2 

1  46 

1 

145 

1 

144 

1 

143 

1 

142 

1 

141 

1 

1  39 

1 

3 

2  39 

2 

237 

2 

236 

2 

234 

2 

233 

2 

231 

2 

229 

2 

4 

332 

3 

330 

4 

328 

4 

326 

4 

323 

4 

321 

4 

3  19 

4 

5 

425 

5 

422 

6 

420 

6 

4  17 

6 

4  14 

6 

4  12 

6 

4  9 

6 

6 

5  18 

8 

5  15 

8 

5  12 

8 

5  8 

8 

5  5 

8 

5  2 

9 

58 

9 

7 

6  11 

11 

6  7 

11 

6  4 

11 

6  0 

11 

56 

11 

52 

12 

548 

12 

8 

7  4 

14 

59 

14 

55 

15 

51 

15 

647 

15 

642 

15 

638 

16 

9 

56 

18 

752 

18 

747 

19 

743 

19 

737 

19 

732 

19 

727 

20 

10 

849 

22 

844 

22 

839 

23 

834 

23 

828 

24 

822 

.24 

8  17 

25 

11 

942 

27 

936 

27 

931 

28 

925 

28 

9  19 

29 

9  12 

29 

9  6 

30 

12 

1035 

32 

1029 

32 

1022 

33 

10  16 

34 

10  9 

34 

10  2 

35 

56 

35 

13 

11  27 

37 

11  21 

38 

11  14 

39 

11  7 

40 

11  0 

40 

52 

41 

1045 

41 

14 

1220 

43 

12  13 

44 

12  6 

45 

58 

46 

50 

47 

1142 

48 

11  34 

48 

15 

13  13 

50 

13  5 

51 

57 

52 

1249 

53 

1241 

54 

1232 

55 

1223 

56 

16 

14  5 

57 

57 

58 

1349 

59 

1340 

32  1 

1331 

33  2 

1322 

34  3 

13  13 

35  4 

17 

58 

29  4 

1449 

30  6 

1440 

31  7 

1431 

9 

1421 

10 

14  12 

11 

14  2 

12 

18 

1550 

12 

1541 

14 

1531 

16 

1522 

17 

15  12 

18 

15  1 

20 

51 

21 

19 

1642 

21 

1633 

23 

1623 

25 

1612 

26 

16  2 

27 

51 

29 

1540 

30 

20 

1735 

30 

1724 

32 

17  14 

34 

17  3 

36 

52 

37 

1640 

39 

1628 

40 

21 

1827 

40 

18  16 

42 

18  5 

44 

53 

46 

1742 

48 

1729 

49 

17  17 

51 

22 

19  19 

50 

19  8 

52 

56 

55 

1844 

57 

1831 

59 

18  19 

35  0 

18  6 

36  2 

23 

20  11 

30  1 

59 

31  3 

1947 

32  6 

1934 

33  8 

1921 

34  10 

19  8 

12 

54 

14 

24 

21  3 

12 

2050 

15 

2037 

18 

2024 

20 

20  11 

22 

57 

24 

1942 

26 

25 

55 

24 

2141 

27 

21  28 

30 

21  14 

33 

21  0 

35 

2046 

37 

2030 

39 

26 

2246 

37 

2232 

40 

22  19 

43 

22  4 

46 

49 

48 

21  34 

51 

21  18 

53 

27 

2338 

50 

2323 

53 

23  9 

57 

54 

34  0 

2238 

35  2 

2223 

36  5 

22  6 

37  8 

28 

2429 

31  3 

24  14 

32  7 

59 

33  11 

2344 

14 

2327 

17 

23  11 

20 

54 

23 

29 

2521 

18 

25  5 

22 

2449 

26 

2433 

29 

24  16 

33 

59 

36 

2342 

38 

30 

2612 

33 

56 

37 

2539 

41 

2523 

45 

25  5 

49 

2447 

52 

2429 

55 

31 

27  3 

49 

2647 

53 

2629 

58 

2612 

35  2 

54 

36  6 

2535 

37  9 

25  16 

3812 

32 

54 

32  5 

2737 

3310 

27  19 

3415 

27  1 

19 

2642 

23 

2623 

27 

26  3 

30 

33 

2845 

22 

2827 

28 

28  9 

33 

50 

37 

2730 

41 

2711 

45 

50 

49 

34 

29  35 

40 

29  17 

46 

58 

51 

2839 

56 

28  18 

37  0 

58 

38  4 

2737 

39  8 

35 

3026 

59 

30  7 

34  5 

2947 

3511 

2927 

3616 

29  6 

20 

2845 

24 

2824 

28 

36 

31  16 

33  19 

56 

25 

3036 

31 

30  15 

36 

54 

41 

2932 

45 

2910 

49 

37 

32  6 

39 

3146 

46 

3125 

52 

31  3 

57 

3041 

38  2 

3019 

39  7 

56 

40  11 

38 

56 

34  1 

3235 

35  7 

32  13 

3614 

51 

3720 

31  28 

25 

31  5 

30 

3042 

34 

39 

3346 

23 

3324 

30 

33  1 

37 

3239 

43 

32  15 

48 

51 

53 

3127 

57 

40 

3435 

46 

34  13 

53 

49 

37  0 

3326 

38  7 

33  2 

39  12 

3237 

40  17 

32  12 

41  22 

41 

3524 

35  10 

35  1 

36  18 

3437 

25 

34  13 

31 

48 

37 

3323 

43 

57 

47 

42 

3613 

35 

49 

43 

3525 

51 

35  0 

57 

3434 

40  4 

34  8 

41  9 

3342 

42  14 

43 

37  2 

36  1 

3637 

37  10 

3612 

3817 

46 

3924 

3520 

31 

53 

36 

3426 

41 

44 

50 

28 

3725 

37 

59 

45 

3633 

52 

36  6 

59 

3538 

42  5 

35  10 

43  10 

45 

3838 

57 

38  12 

38  5 

3746 

3914 

3719 

4021 

51 

4128 

3622 

34 

53 

39 

Table  13.    Kelvin's  Sumner  Line  Table 


301 


b 

a  =  28° 

a  =  29° 

a  =  30° 

a  =  31° 

a  =  32° 

a  =  33° 

a  =  34° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

3838 

3657 

38  12 

38  5 

3746 

39  14 

37  19 

4021 

3651 

41  28 

3622 

4234 

3553 

43  39 

46 

3926 

3726 

59 

35 

3832 

44 

38  4 

52 

3736 

58 

37  6 

43  4 

3636 

44  9 

47 

40  13 

56 

3946 

39  6 

39  18 

40  15 

49 

41  23 

3820 

4230 

50 

36 

37  19 

41 

48 

41  0 

3828 

4032 

38 

40  4 

47 

3934 

55 

39  4 

43  2 

3833 

44  9 

38  2 

45  14 

49 

47 

39  1 

41  18 

4011 

49 

4121 

40  19 

4229 

48 

36 

39  16 

43 

44 

48 

50 

4234 

36 

42  4 

46 

41  34 

56 

41  3 

43  4 

4031 

4411 

59 

45  18 

3926 

4623 

51 

4320 

40  12 

49 

41  22 

42  18 

4232 

46 

40 

41  14 

48 

4041 

54 

40  7 

59 

52 

44  5 

49 

4334 

42  0 

43  2 

4310 

4229 

44  18 

56 

4526 

41  22 

4632 

48 

4737 

53 

50 

41  28 

44  18 

39 

46 

49 

43  12 

57 

4238 

46  5 

42  3 

47  11 

41  28 

48  16 

54 

4535 

42  8 

45  2 

43  19 

4429 

4429 

54 

4538 

43  19 

45 

44 

51 

42  7 

56 

55 

46  19 

50 

46 

44  1 

4511 

4511 

4436 

4620 

44  0 

4727 

4324 

4833 

46 

4937 

56 

47  3 

4334 

4629 

45 

53 

55 

45  17 

47  4 

40 

4810 

44  3 

49  16 

4325 

5020 

57 

46 

44  19 

47  11 

4530 

4635 

4640 

58 

49 

4520 

55 

42 

50  1 

44  3 

51  5 

58 

4829 

45  6 

53 

46  17 

47  16 

4727 

4638 

4835 

59 

4942 

4520 

47 

40 

51 

59 

49  11 

55 

4834 

47  6 

56 

48  16 

47  17 

4924 

4638 

5030 

58 

51  35 

45  17 

52  38 

60 

52 

4646 

49  14 

57 

4836 

49  6 

56 

50  14 

4716 

5120 

4635 

5224 

53 

5327 

61 

5033 

4738 

54 

4850 

49  15 

59 

4834 

51  6 

53 

52  12 

47  11 

53  15 

4628 

54  18 

62 

51  13 

4833 

5033 

4944 

53 

5053 

49  11 

52  0 

4829 

53  5 

46 

54  8 

47  3 

55  10 

63 

53 

4930 

51  12 

5041 

5030 

51  49 

48 

56 

49  5 

54  0 

4821 

55  3 

37 

56  4 

64 

5231 

5030 

49 

5140 

51  7 

5248 

5024 

5353 

40 

57 

55 

59 

4810 

59 

65 

53  9 

51  31 

5226 

5241 

43 

5348 

59 

5453 

5014 

5556 

4928 

5657 

42 

5756 

66 

46 

5235 

53  2 

5344 

52  18 

5450 

5133 

5555 

47 

5656 

50  1 

5757 

49  14 

5855 

67 

5422 

5341 

37 

5449 

52 

5555 

52  6 

5658 

51  19 

5759 

32 

5858 

44 

5955 

68 

57 

5450 

5411 

5557 

5325 

57  1 

38 

58  3 

50 

59  4 

51  2 

60  1 

5014 

6057 

69 

5531 

56  1 

44 

57  7 

57 

58  10 

53  9 

59  11 

5221 

6010 

32 

61  6 

43 

62  1 

70 

56  4 

57  15 

55  16 

5819 

5428 

5921 

39 

6021 

50 

61  18 

52  1 

62  13 

51  10 

63  7 

71 

36 

5831 

47 

5934 

58 

6035 

54  8 

6133 

5318 

6229 

28 

6322 

37 

64  14 

72 

57  7 

5950 

5617 

6052 

5527 

61  51 

36 

6247 

45 

6341 

54 

6433 

52  3 

65  23 

73 

36 

61  12 

46 

62  12 

55 

63  9 

55  3 

64  3 

54  11 

6456 

53  19 

6546 

27 

6634 

74 

58  4 

6236 

5713 

6334 

5621 

6429 

29 

6521 

36 

66  12 

43 

67  0 

50 

6746 

75 

31 

64  3 

39 

6458 

46 

6551 

53 

6642 

55  0 

6730 

54  6 

6816 

53  12 

69  0 

76 

57 

6532 

58  4 

6625 

5710 

6716 

5616 

68  4 

22 

6850 

28 

6934 

33 

70  16 

77 

5921 

67  4 

27 

6754 

33 

6843 

38 

6929 

43 

7012 

48 

7054 

53 

71  33 

78 

44 

6839 

49 

6926 

54 

7012 

58 

7055 

56  3 

7136 

55  7 

72  15 

54  11 

7252 

79 

60  5 

7016 

59  9 

71  0 

5813 

7143 

5717 

7223 

21 

73  1 

25 

7338 

28 

74  12 

80 

24 

7155 

28 

7236 

31 

73  16 

35 

7353 

38 

7428 

41 

75  2 

44 

7534 

81 

42 

7336 

45 

74  14 

48 

7450 

51 

7524 

53 

7557 

56 

7627 

58 

7657 

82 

58 

7520 

60  0 

7554 

59  3 

7627 

58  5 

7657 

57  7 

7727 

56  9 

7754 

55  11 

78  21 

83 

61  12 

77  6 

14 

7736 

16 

78  5 

18 

7832 

19 

7858 

21 

7922 

22 

7946 

84 

25 

7853 

26 

79  19 

28 

7944 

29 

80  8 

30 

8030 

31 

8051 

32 

81  12 

85 

36 

8042 

36 

81  4 

38 

8125 

38 

8145 

39 

82  4 

40 

822-1 

41 

82  38 

86 

44 

8232 

45 

8250 

46 

83  7 

46 

8323 

47 

8338 

47 

8352 

48 

84  6 

87 

51 

8423 

51 

8436 

52 

8449 

52 

85  1 

53 

8513 

53 

8523 

53 

85  34 

88 

56 

86  15 

56 

8624 

56 

8632 

56 

8640 

57 

8648 

57 

8655 

57 

87  2 

89 

59 

88  7 

59 

88  12 

59 

8816 

59 

8820 

59 

8824 

59 

8827 

59 

8831 

90 

62  0 

90  0 

61  0 

90  0 

60  0 

90  0 

59  0 

90  0 

58  0 

90  0 

57  0 

90  0 

56  0 

90  0 

302 


Table  13.    Kelvin's  Simmer  Line  Table 


b 

a  =  35° 

a  =  36° 

a  =  37° 

a  =  38° 

a  =  39° 

a  =  40° 

a  =  41° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o   f 

0 

0  0 

35  0 

0  0 

36  0 

0  0 

37  0 

0  0 

38  0 

0  0 

39  0 

0  0 

40  0 

0  0 

41  0 

1 

49 

0 

49 

0 

48 

0 

47 

0 

47 

0 

46 

0 

45 

0 

2 

1  38 

1 

137 

1 

136 

1 

135 

1 

133 

1 

132 

131 

1 

3 

227 

2 

226 

2 

224 

2 

222 

ey 

220 

f 

218 

f 

216 

2 

4 

317 

4 

3  14 

4 

3  12 

4 

3  9 

4 

3  7 

4 

3  4 

4 

3  1 

4 

5 

4  6 

6 

4  3 

6 

59 

6 

56 

6 

53 

6 

50 

6 

46 

6 

6 

55 

9 

51 

g 

447 

9 

444 

9 

440 

9 

436 

9 

431 

9 

7 

544 

12 

539 

12 

535 

12 

531 

12 

526 

13 

521 

13 

5  17 

13 

8 

633 

16 

628 

16 

623 

16 

618 

16 

613 

17 

6  7 

17 

6  2 

17 

9 

722 

20 

716 

20 

7  11 

20 

7  5 

21 

59 

21 

53 

21 

47 

21 

10 

811 

25 

8  5 

25 

58 

25 

52 

26 

745 

26 

739 

26 

732 

26 

11 

9  0 

30 

53 

30 

846 

31 

839 

31 

832 

31 

824 

31 

8  17 

32 

12 

48 

36 

941 

36 

934 

37 

926 

37 

918 

37 

9  10 

37 

9  2 

38 

13 

1037 

42 

1029 

43 

1021 

43 

1013 

43 

10  4 

44 

55 

44 

47 

44 

14 

1126 

49 

11  17 

50 

11  8 

50 

59 

50 

50 

51 

1041 

51 

1031 

51 

15 

12  14 

56 

12  5 

57 

56 

57 

11  46 

58 

11  36 

59 

11  26 

59 

11  16 

59 

16 

13  3 

36  4 

53 

37  5 

1243 

38  5 

1233 

39  6 

1222 

40  7 

12  11 

41  7 

12  0 

42  7 

17 

51 

13 

1341 

13 

1330 

14 

13  19 

15 

13  8 

16 

56 

16 

45 

16 

18 

1440 

22 

1429 

22 

14  17 

23 

14  6 

24 

54 

25 

1341 

25 

1329 

26 

19 

1528 

31 

1516 

32 

15  4 

33 

52 

34 

1440 

35 

1426 

35 

14  13 

36 

20 

16  16 

41 

16  4 

43 

51 

44 

1538 

44 

1525 

45 

15  11 

46 

57 

46 

21 

17  4 

52 

51 

54 

1638 

55 

1624 

55 

16  10 

56 

56 

57 

1541 

57 

22 

52 

37  4 

1738 

38  5 

1725 

39  6 

17  10 

40  7 

55 

41  8 

1641 

42  9 

1625 

43  9 

23 

1840 

16 

1825 

17 

18  11 

18 

56 

19 

1740 

20 

1725 

21 

17  9 

22 

24 

1928 

28 

1912 

30 

57 

31 

1842 

32 

1825 

33 

18  9 

34 

53 

35 

25 

20  15 

41 

59 

43 

19  43 

45 

1927 

46 

19  10 

47 

53 

48 

1836 

48 

26 

21  3 

55 

2046 

57 

20  29 

59 

20  13 

41  0 

55 

42  1 

1937 

43  2 

19  19 

44  3 

27 

50 

3810 

2133 

39  12 

21  15 

40  13 

58 

15 

2040 

16 

2021 

17 

20  2 

18 

28 

2237 

25 

22  19 

27 

22  1 

29 

2143 

30 

21  24 

32 

21  5 

33 

45 

33 

29 

2324 

41 

23  5 

43 

47 

45 

2228 

46 

22  8 

48 

48 

49 

2128 

49 

30 

2411 

57 

51 

40  0 

23  32 

41  2 

2312 

42  3 

52 

43  5 

2231 

44  6 

2210 

45  6 

31 

57 

3915 

2437 

17 

24  17 

19 

57 

21 

2336 

22 

23  14 

23 

52 

24 

32 

2544 

33 

2523 

35 

25  2 

37 

2441 

39 

2419 

41 

57 

42 

2334 

43 

33 

2630 

52 

26  9 

54 

47 

56 

2525 

58 

25  2 

44  0 

2440 

45  1 

24  16 

46  2 

34 

2716 

40  11 

54 

41  14 

2632 

42  16 

26  9 

43  18 

45 

20 

2522 

21 

58 

22 

35 

28  2 

31 

2739 

34 

27  16 

37 

52 

39 

2628 

40 

26  4 

41 

2539 

42 

36 

47 

52 

2824 

56 

28  0 

58 

2735 

44  0 

2711 

45  2 

46 

46  3 

2620 

47  3 

37 

2932 

41  14 

29  8 

42  18 

44 

4320 

2818 

22 

53 

24 

2727 

25 

27  1 

25 

38 

3017 

37 

52 

41 

2927 

43 

29  1 

45 

2835 

47 

28  8 

48 

41 

48 

39 

31  2 

42  1 

3036 

43  4 

3010 

44  7 

44 

45  9 

2917 

46  11 

49 

47  12 

2821 

48  12 

40 

46 

26 

31  20 

29 

53 

32 

3026 

34 

58 

35 

2930 

36 

29  1 

37 

41 

3230 

51 

32  3 

55 

31  36 

58 

31  8 

46  0 

3039 

47  1 

3010 

48  2 

41 

49  2 

42 

33  14 

43  18 

46 

4421 

32  18 

4524 

49 

26 

31  20 

•  28 

50 

28 

3020 

28 

43 

58 

45 

3329 

49 

33  0 

52 

3230 

53 

32  0 

55 

31.30 

55 

59 

55 

44 

3441 

44  14 

34  12 

45  17 

42 

4620 

33  11 

4722 

40 

4823 

32  9 

4924 

3137 

5023 

45 

3524 

43 

54 

46 

3423 

49 

52 

51 

3320 

52 

48 

53 

32  15 

52 

Table  13.    Kelvin's  Simmer  Line  Table 


303 


b 

a  =35° 

a  =  36° 

a  =  37° 

a  =  38° 

a  =  39° 

a  =  40° 

a  =  41° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

3524 

4443 

3454 

4546 

3423 

4649 

3352 

4751 

3320 

4852 

3248 

4953 

32  15 

5052 

46 

36  6 

4514 

3535 

46  17 

35  4 

4720 

3432 

4822 

59 

4923 

3326 

5023 

53 

5122 

47 

48 

45 

3616 

49 

44 

51 

3512 

53 

3438 

54 

34  4 

54 

3330 

53 

48 

3730 

4618 

57 

4721 

3624 

4824 

51 

4925 

3517 

5026 

42 

5126 

34  7 

5225 

49 

3811 

52 

3738 

55 

37  4 

57 

3630 

59 

55 

59 

35  19 

59 

43 

57 

50 

52 

4727 

38  18 

4830 

43 

4932 

37  8 

5033 

3632 

5133 

56 

5233 

3519 

5331 

51 

3932 

48  3 

57 

49  6 

3822 

50  8 

46 

51  9 

37  9 

52  9 

3632 

53  8 

55 

54  6 

52 

4012 

41 

3936 

43 

39  0 

45 

3823 

46 

46 

45 

37  8 

44 

3630 

42 

53 

52 

49  19 

40  15 

5022 

38 

5123 

39  0 

5224 

3822 

5323 

43 

5421 

37  4 

55  18 

54 

41  31 

59 

53 

51  2 

4015 

52  3 

36 

53  3 

57 

54  2 

3818 

59 

38 

56 

55 

42  9 

5041 

4130 

43 

52 

43 

40  12 

43 

3932 

41 

52 

5539 

38  11 

5635 

56 

47 

51  23 

42  7 

5225 

41  28 

5325 

47 

5424 

40  7 

5522 

3926 

56  19 

44 

57  15 

57 

4324 

52  7 

43 

53  9 

42  3 

54  8 

4122 

55  7 

41 

56  4 

59 

57  1 

3916 

56 

58 

44  0 

53 

43  19 

54 

38 

53 

56 

51 

41  14 

48 

4031 

44 

48 

5838 

59 

36 

5340 

54 

5440 

43  12 

5539 

4229 

5636 

46 

5733 

41  3 

5828 

4019 

5921 

60 

4511 

5428 

4429 

5528 

46 

5626 

43  2 

5723 

42  18 

5819 

34 

5913 

49 

60  6 

61 

46 

5518 

45  2 

56  17 

44  19 

5715 

34 

58  11 

49 

59  6 

42  4 

59 

41  18 

51 

62 

1620 

56  10 

35 

57  8 

51 

58  5 

44  5 

59  0 

4320 

54 

34 

6047 

47 

6138 

63 

53 

57  3 

46  7 

58  0 

4522 

56 

36 

50 

50 

6043 

43  3 

61  35 

42  15 

6225 

64 

4725 

57 

39 

54 

52 

5949 

45  6 

6042 

4419 

6134 

31 

6225 

43 

63  14 

65 

56 

5853 

47  9 

5949 

4622 

6043 

35 

61  35 

47 

6226 

58 

6316 

43  9 

64  4 

66 

4827 

5951 

39 

6046 

51 

61  39 

46  3 

6230 

45  14 

6320 

4425 

64  8 

35 

55 

67 

56 

6050 

48  8 

61  44 

47  19 

6236 

30 

6326 

40 

64  15 

51 

65  2 

44  0 

6548 

68 

4925 

61  51 

36 

6244 

46 

6334 

56 

6423 

46  6 

6511 

4516 

57 

24 

6641 

69 

53 

6254 

49  3 

6345 

4813 

6434 

4722 

6522 

31 

66  8 

40 

6653 

48 

6736 

70 

5020 

6358 

29 

6448 

38 

6535 

46 

6622 

55 

67  6 

46  3 

6750 

45  10 

6831 

71 

46 

65  4 

54 

6552 

49  2 

6638 

4810 

6723 

47  17 

68  6 

25 

6848 

32 

6928 

72 

51  10 

66  11 

50  18 

6658 

26 

6742 

33 

6825 

39 

69  7 

46 

6947 

52 

7026 

73 

34 

6720 

41 

68  5 

48 

6848 

54 

6929 

48  0 

70  9 

47  6 

7047 

4612 

7125 

74 

57 

6831 

51  3 

69  14 

50  9 

6955 

49  15 

7034 

20 

71  12 

26 

7149 

30 

7225 

75 

5218 

6943 

24 

7024 

29 

71  3 

34 

7140 

39 

7216 

44 

7251 

48 

7325 

76 

38 

7056 

43 

7135 

48 

72  12 

52 

7248 

57 

7322 

48  1 

7355 

47  5 

7427 

77 

57 

72  11 

52  1 

7248 

51  6 

7323 

50  9 

7356 

49  13 

7429 

17 

75  0 

20 

7530 

78 

53  15 

7328 

18 

74  2 

22 

7435 

25 

75  6 

29 

7536 

32 

76  5 

35 

7633 

79 

31 

7445 

34 

75  17 

37 

7547 

40 

7617 

43 

7645 

46 

77  11 

48 

7737 

80 

46 

76  4 

49 

7633 

51 

77  1 

54 

7728 

56 

7754 

58 

7818 

8  0 

7842 

81 

54  0 

7724 

53  2 

7751 

52  4 

78  16 

51  6 

7841 

0  8 

79  4 

4910 

7926 

12 

7948 

82 

13 

7846 

14 

79  9 

16 

7932 

17 

7954 

19 

SO  15 

20 

8035 

22 

8054 

83 

24 

80  8 

25 

8029 

26 

8049 

27 

81  8 

29 

81  27 

29 

8144 

31 

82  1 

84 

33 

81  31 

34 

8149 

35 

82  6 

36 

8223 

37 

8239 

37 

8254 

39 

83  9 

85 

41 

8254 

42 

83  10 

43 

8324 

43 

8338 

44 

8352 

44 

84  4 

45 

8417 

86 

48 

84  19 

49 

8431 

49 

8443 

49 

8454 

50 

85  5 

50 

85  15 

50 

8525 

87 

53 

8544 

54 

8553 

54 

86  2 

54 

8610 

54 

86  18 

54 

8626 

54 

8633 

88 

57 

87  9 

57 

8715 

57 

8721 

57 

8726 

57 

8732 

57 

8737 

57 

8742 

89 

59 

8834 

59 

8837 

59 

8840 

59 

8843 

59 

8846 

59 

8848 

59 

8851 

90 

55  0 

90  0 

54  0 

90  0 

53  0 

90  0 

52  0 

90  0 

1  0 

90  0 

0  0 

0  0 

9  0 

90  0 

304 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  =  42° 

a  =  43° 

a  =  44° 

a  =  45° 

a  =  46° 

a  =  47° 

a  =  48° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

42  0 

0  0 

43  0 

0  0 

44  0 

0  0 

45  0 

0  0 

46  0 

0  0 

47  0 

0  0 

48  0 

1 

45 

0 

44 

0 

43 

0 

42 

0 

42 

0 

41 

0 

40 

0 

2 

129 

1 

1  28 

1 

1  26 

1 

1  25 

1 

1  23 

1 

1  22 

1 

1  20 

1 

<! 

214 

2 

2  12 

o 

2  10 

<y 

2  7 

2 

2  5 

2 

2  3 

2 

2  0 

2 

4 

58 

4 

55 

4 

53 

4 

50 

4 

47 

4 

1  44 

4 

40 

4 

5 

343 

6 

339 

6 

336 

6 

332 

6 

328 

6 

325 

6 

320 

6 

6 

427 

'  9 

423 

g 

4  19 

9 

4  14 

9 

410 

9 

4  5 

9 

4  0 

9 

7 

512 

13 

5  7 

13 

5  2 

13 

57 

13 

51 

13 

46 

13 

40 

13 

8 

56 

17 

51 

17 

45 

17 

539 

17 

533 

17 

527 

17 

520 

17 

9 

641 

21 

634 

21 

628 

21 

621 

21 

614 

21 

6  7 

21 

6  0 

21 

10 

725 

26 

718 

26 

711 

26 

7  3 

26 

56 

26 

48 

26 

40 

26 

11 

8  9 

32 

8  1 

32 

53 

32 

45 

32 

737 

32 

729 

32 

720 

32 

12 

53 

38 

45 

38 

836 

38 

827 

38 

818 

38 

8  9 

38 

8  0 

38 

13 

937 

45 

928 

45 

9  19 

45 

9  9 

45 

59 

45 

49 

45 

39 

44 

14 

1021 

52 

10  11 

52 

10  1 

52 

51 

52 

940 

52 

930 

52 

919 

51 

15 

11  5 

59 

55 

44  0 

44 

45  0 

1033 

46  0 

1021 

47  0 

10  10 

48  0 

58 

59 

16 

49 

43  8 

1138 

8 

1126 

8 

11  15 

8 

11  2 

8 

50 

8 

1038 

49  7 

17 

1233 

17 

1221 

17 

12  8 

17 

56 

17 

43 

17 

11  30 

17 

11  17 

16 

18 

13  17 

26 

13  4 

26 

50 

26 

1237 

26 

1224 

26 

12  10 

26 

56 

25 

19 

14  0 

36 

46 

36 

1332 

36 

1318 

36 

13  4 

36 

50 

36 

1235 

35 

20 

44 

47 

1429 

47 

14  14 

47 

59 

47 

45 

47 

1329 

46 

13  14 

46 

21 

1527 

58 

1512 

58 

56 

58 

1440 

58 

1425 

58 

14  9 

57 

53 

57 

22 

16  10 

44  10 

54 

4510 

1538 

46  10 

1521 

47  10 

15  5 

48  10 

48 

49  9 

1431 

50  9 

23 

53 

22 

1636 

22 

1619 

22 

16  2 

22 

45 

22 

1527 

21 

15  9 

21 

24 

1736 

35 

17  18 

35 

17  1 

35 

43 

35 

1625 

35 

16  6 

34 

47 

34 

25 

1819 

49 

18  0 

49 

42 

49 

1723 

49 

17  5 

49 

45 

48 

1625 

47 

26 

19  1 

45  3 

42 

46  3 

1823 

47  3 

18  3 

48  3 

44 

49  3 

1724 

50  2 

17  3 

51  1 

27 

43 

18 

19  24 

18 

19  4 

18 

43 

18 

1823 

17 

18  2 

17 

41 

16 

28 

2025 

34 

20  5 

34 

44 

34 

1923 

33 

19  2 

33 

40 

32 

18  19 

31 

29 

21  7 

50 

46 

50 

2025 

50 

20  3 

49 

41 

49 

19  18 

48 

56 

47 

30 

49 

46  7 

21  27 

47  7 

21  5 

48  7 

42 

49  6 

2020 

50  6 

56 

51  5 

1933 

52  3 

31 

2230 

25 

22  8 

25 

45 

25 

21  21 

24 

58 

23 

2034 

22 

2010 

20 

32 

23  11 

43 

48 

43 

2225 

43 

22  0 

42 

21  36 

41 

21  11 

40 

46 

38 

33 

52 

47  2 

2328 

48  2 

23  4 

49  2 

39 

50  1 

22  14 

51  0 

48 

58 

2122 

56 

34 

2433 

22 

24  8 

22 

43 

21 

23  18 

20 

52 

19 

2225 

52  17 

58 

5315 

35 

2514 

42 

48 

42 

2422 

42 

56 

41 

2329 

39 

23  2 

37 

2234 

35 

36 

54 

48  4 

2528 

49  3 

25  1 

50  3 

2434 

51  2 

24  6 

52  0 

38 

58 

23  10 

56 

37 

2634 

26 

26  7 

25 

39 

25 

25  11 

23 

43 

22 

24  14 

53  19 

45 

54  17 

38 

27  14 

49 

46 

48 

2617 

47 

48 

46 

25  19 

44 

50 

41 

2420 

39 

39 

53 

49  12 

2724 

50  12 

55 

51  11 

2625 

52  9 

55 

53  7 

2525 

54  4 

54 

55  1 

40 

2832 

37 

28  2 

36 

2732 

35 

27  2 

33 

2631 

31 

26  0 

28 

2528 

24 

41 

29  11 

50  2 

40 

51  1 

28  9 

52  0 

38 

58 

27  7 

55 

35 

52 

26  2 

48 

42 

49 

28 

2918 

27 

46 

25 

2814 

5323 

42 

5420 

27  9 

55  17 

36 

56  13 

43 

3027 

55 

55 

54 

2923 

52 

50 

49 

2817 

46 

43 

42 

27  9 

38 

44 

31  5 

51  23 

3032 

5221 

59 

53  19 

2925 

54  16 

51 

5513 

28  17 

56  9 

42 

57  4 

45 

42 

51 

31  9 

50 

3034 

47 

30  0 

44 

2925 

40 

50 

36 

2814 

31 

Table  13.    Kelvin's  Sumner  Line  Table 


305 


b 

a  =  43° 

a  =  43° 

a  =  44° 

a  =  45° 

a  =  46° 

a  =  47° 

a  =  48° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

3142 

51  51 

31  9 

5250 

3034 

5347 

30  0 

5444 

2925 

5540 

2850 

5636 

28  14 

5731 

46 

32  19 

5221 

45 

53  19 

31  10 

54  16 

34 

55  13 

59 

56  9 

2923 

57  4 

46 

59 

47 

55 

52 

3220 

49 

45 

46 

31  8 

42 

3032 

38 

55 

33 

29  18 

5827 

48 

3331 

5323 

55 

5420 

32  19 

55  17 

42 

56  13 

31  5 

57  8 

3027 

58  2 

49 

56 

49 

34  7 

55 

3330 

52 

53 

49 

32  15 

44 

37 

39 

59 

33 

3020 

5926 

50 

42 

5429 

34  4 

5525 

3326 

5621 

48 

57  16 

32  9 

5810 

3130 

59  4 

50 

56 

51 

3517 

55  3 

38 

59 

59 

55 

3320 

49 

40 

43 

32  0 

36 

31  20 

6028 

52 

51 

38 

3511 

5634 

3432 

5729 

52 

5823 

33  11 

59  16 

30 

60  8 

49 

61  0 

53 

3624 

5615 

44 

5710 

35  4 

58  4 

3423 

58 

42 

50 

33  0 

42 

32  18 

33 

54 

57 

52 

3617 

47 

35 

40 

54 

5933 

3412 

6025 

29 

61  16 

46 

62  7 

55 

3730 

5730 

48 

5824 

36  6 

5917 

3524 

6010 

41 

61  1 

58 

52 

33  14 

41 

56 

38  2 

58  9 

3719 

59  3 

36 

56 

53 

47 

3510 

38 

3426 

6228 

41 

63  17 

57 

33 

50 

50 

43 

37  6 

6035 

3622 

61  26 

38 

62  15 

53 

63  4 

34  8 

53 

58 

39  4 

5931 

3820 

6024 

35 

61  15 

51 

62  5 

36  6 

54 

3520 

42 

34 

6430 

59 

34 

60  14 

49 

61  5 

38  4 

56 

37  19 

45 

33 

6333 

46 

6421 

35  0 

65  7 

60 

40  4 

57 

3918 

48 

32 

6238 

46 

6326 

59 

6414 

36  12 

65  0 

25 

46 

61 

33 

6142 

46 

6232 

59 

6321 

3812 

64  8 

3725 

55 

37 

40 

49 

6625 

62 

41  1 

6228 

40  13 

63  17 

3926 

64  4 

38 

51 

50 

6537 

37  1 

6621 

3613 

67  5 

63 

28 

63  15 

40 

64  2 

52 

49 

39  3 

6535 

38  14 

6620 

25 

67  3 

36 

46 

64 

54 

64  2 

41  6 

49 

40  17 

6535 

27 

6620 

38 

67  3 

48 

46 

58 

6828 

65 

4220 

51 

31 

6537 

41 

6622 

51 

67  5 

39  1 

48 

3810 

6830 

3720 

69  10 

66 

45 

6541 

55 

6626 

41  5 

67  10 

40  14 

52 

23 

6833 

32 

69  14 

41 

53 

67 

43  10 

6632 

42  19 

67  16 

28 

58 

36 

6840 

45 

6920 

53 

59 

38  1 

7037 

68 

33 

6725 

42 

68  7 

50 

6848 

58 

6928 

40  6 

70  7 

3913 

7045 

21 

7122 

69 

56 

6818 

43  4 

59 

42  11 

6938 

41  19 

7017 

26 

55 

33 

7131 

40 

72  7 

70 

44  18 

69  12 

25 

6952 

31 

7030 

39 

71  7 

45 

7143 

51 

72  19 

58 

53 

71 

39 

70  7 

45 

7045 

51 

71  22 

58 

58 

41  3 

7233 

40  9 

73  7 

3915 

7340 

72 

59 

71  3 

44  4 

7140 

43  10 

72  15 

42  16 

7250 

21 

7323 

26 

56 

31 

7427 

73 

45  18 

72  1 

22 

7236 

28 

73  9 

33 

7342 

38 

74  14 

42 

7445 

47 

75  15 

74 

36 

59 

40 

7332 

45 

74  4 

49 

7435 

54 

75  6 

58 

7535 

40  2 

76  4 

75 

53 

7358 

57 

7429 

44  1 

75  0 

43  5 

7529 

42  9 

58 

41  12 

7626 

16 

53 

76 

46  9 

7458 

45  12 

7527 

16 

56 

19 

7624 

23 

7651 

26 

77  17 

29 

7743 

77 

24 

7558 

27 

7626 

30 

7653 

33 

77  19 

36 

7745 

39 

78  9 

41 

7833 

78 

38 

77  0 

41 

7726 

43 

7751 

46 

7815 

48 

7839 

51 

79  2 

53 

7924 

79 

51 

78  2 

53 

7826 

55 

7849 

57 

79  12 

43  0 

7934 

42  2 

55 

41  3 

8015 

80 

47  3 

79  5 

46  5 

7927 

45  6 

7948 

44  8 

80  9 

10 

8029 

12 

8048 

13 

81  7 

81 

13 

80  9 

15 

8029 

16 

8048 

18 

81  7 

20 

81  25 

21 

8142 

22 

59 

82 

23 

81  13 

24 

81  31 

26 

8148 

27 

82  5 

28 

8221 

29 

8236 

30 

8252 

83 

32 

82  18 

32 

8233 

34 

8248 

34 

83  3 

35 

83  17 

36 

8331 

37 

8345 

84 

39 

8323 

40 

8336 

41 

8349 

41 

84  2 

42 

84  14 

42 

8426 

43 

8438 

85 

46 

8428 

46 

8439 

47 

8450 

47 

85  1 

47 

8511 

48 

8521 

48 

8531 

86 

51 

8534 

51 

8543 

51 

8552 

52 

86  0 

52 

86  9 

52 

86  17 

52 

8624 

87 

55 

8640 

55 

8647 

55 

8654 

55 

87  0 

55 

87  6 

56 

87  12 

56 

8718 

88 

58 

8747 

58 

8751 

58 

8756 

58 

88  0 

58 

88  4 

58 

88  8 

58 

8812 

89 

59 

8853 

59 

8856 

59 

8858 

59 

89  0 

59 

89  2 

43  0 

89  4 

42  0 

89  6 

90 

48  0 

90  0 

47  0 

90  0 

46  0 

90  0 

45  0 

90  0 

44  0 

90  0 

0 

90  0 

0 

90  0 

306 


Table  13.    Kelvin's  Simmer  Line  Table 


b 

a  =  49° 

a  =  50° 

a  =  51° 

a  =  52° 

a  =  53° 

a  =  54° 

a  =  55° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

49  0 

0  0 

50  0 

0  0 

51  0 

0  0 

52  0 

0  0 

53  0 

0  0 

54  0 

0  0 

55  0 

1 

39 

0 

39 

0 

38 

0 

37 

0 

36 

0 

35 

0 

34 

0 

2 

1  19 

1 

1  17 

1 

1  16 

1 

1  14 

1 

1  12 

1 

1  11 

1 

1  9 

1 

3 

58 

2 

56 

2 

53 

2 

51 

2 

48 

2 

46 

2 

43 

2 

4 

237 

4 

234 

4 

231 

4 

228 

4 

224 

4 

221 

4 

2  18 

4 

5 

317 

6 

313 

6 

3  9 

6 

3  5 

6 

3  0 

6 

56 

6 

52 

6 

6 

56 

9 

51 

9 

47 

9 

41 

9 

36 

9 

331 

9 

326 

9 

7 

435 

13 

430 

13 

424 

13 

418 

12 

412 

12 

4  6 

12 

4  0 

12 

8 

514 

17 

5  8 

17 

5  2 

17 

55 

16 

48 

16 

41 

16 

35 

16 

9 

53 

21 

46 

21 

39 

21 

532 

21 

524 

20 

5  16 

20 

5  9 

20 

10 

632 

26 

625 

26 

616 

26 

6  8 

26 

6  0 

25 

51 

25 

43 

25 

11 

711 

32 

7  3 

31 

54 

31 

45 

31 

36 

30 

626 

30 

617 

30 

12 

50 

38 

41 

37 

731 

37 

721 

37 

711 

36 

7  1 

36 

51 

36 

13 

829 

44 

8  19 

44 

8  8 

44 

58 

43 

47 

43 

36 

42 

725 

42 

14 

9  8 

51 

57 

51 

45 

51 

834 

50 

822 

50 

811 

49 

59 

49 

15 

47 

59 

935 

59 

922 

58 

910 

58 

58 

57 

45 

56 

832 

56 

16 

1025 

50  7 

10  12 

51  7 

59 

52  6 

46 

53  6 

933 

54  5 

919 

55  4 

9  6 

56  3 

17 

11  4 

16 

50 

15 

1036 

15 

1022 

14 

10  8 

13 

54 

12 

39 

11 

18 

42 

25 

11  28 

24 

11  13 

24 

58 

23 

43 

22 

1028 

21 

1013 

20 

19 

1220 

35 

12  5 

34 

49 

34 

11  34 

33 

11  18 

32 

11  2 

31 

46 

29 

20 

58 

45 

42 

45 

1226 

44 

12  9 

43 

53 

42 

36 

41 

11  19 

39 

21 

1336 

56 

13  19 

56 

13  2 

55 

45 

54 

1227 

52 

12  10 

51 

52 

49 

22 

14  14 

51  8 

56 

52  7 

38 

53  6 

1320 

54  5 

13  2 

55  3 

43 

56  2 

1225 

57  0 

23 

51 

20 

1433 

19 

14  14 

18 

55 

17 

36 

15 

13  17 

14 

57 

12 

24 

1529 

33 

15  9 

32 

50 

30 

1430 

29 

14  10 

27 

50 

26 

1330 

24 

25 

16  6 

46 

46 

45 

1526 

43 

15  5 

42 

44 

40 

1423 

38 

14  2 

36 

26 

43 

52  0 

1622 

59 

16  1 

57 

40 

55 

1518 

53 

56 

51 

34 

49 

27 

1720 

14 

58 

53  13 

36 

5411 

16  14 

55  9 

51 

56  7 

1529 

57  5 

15  6 

58  2 

28 

56 

29 

1734 

28 

1711 

26 

48 

24 

1625 

22 

16  1 

19 

37 

16 

29 

1833 

45 

18  10 

44 

46 

42 

1722 

39 

58 

37 

33 

34 

16  9 

31 

30 

19  9 

53  2 

45 

54  0 

1820 

58 

56 

55 

1731 

52 

17'  5 

49 

40 

46 

31 

45 

19 

1920 

17 

55 

5514 

1829 

5611 

18  3 

57  8 

37 

58  5 

1711 

59  2 

32 

2021 

36 

55 

34 

1929 

31 

19  2 

28 

36 

25 

18  9 

22 

42 

18 

33 

56 

54 

2030 

52 

20  3 

49 

35 

46 

19  8 

42 

40 

39 

1812 

35 

34 

21  31 

54  13 

21  4 

55  11 

37 

56  7 

20  8 

57  4 

40 

58  0 

19  11 

56 

42 

52 

35 

22  6 

33 

38 

30 

21  10 

26 

41 

23 

20  12 

19 

42 

5914 

19  12 

60  10 

36 

41 

53 

22  12 

50 

43 

46 

21  13 

42 

43 

38 

2013 

33 

42 

'28 

37 

23  15 

55  14 

46 

5610 

2216 

57  7 

45 

58  2 

21  14 

58 

43 

52 

2012 

47 

38 

49 

35 

23  19 

32 

48 

28 

2217 

23 

45 

59  18 

21  13 

6012 

41 

61  7 

39 

2423 

57 

52 

54 

2320 

49 

48 

44 

22  15 

39 

43 

33 

21  10 

27 

40 

57 

5620 

2424 

5716 

52 

5811 

23  19 

59  6 

45 

60  0 

2212 

54 

38 

48 

41 

2530 

44 

56 

39 

2423 

34 

50 

29 

23  15 

22 

41 

61  16 

22  6 

62  9 

42 

26  3 

57  8 

2528 

58  3 

54 

58 

2420 

52 

45 

45 

2310 

38 

34 

31 

43 

35 

33 

26  0 

28 

2525 

5922 

50 

6015 

24  14 

61  8 

38 

62  1 

23  2 

53 

44 

27  7 

59 

31 

53 

55 

47 

2519 

40 

43 

32 

24  6 

25 

29 

63  16 

45 

38 

5825 

27  2 

5919 

2625 

6012 

48 

61  5 

2511 

57 

34 

49 

56 

40 

Table  13.    Kelvin's  Sumiier  Line  Table 


307 


b 

a  =  49° 

a  =  50° 

a  =  51° 

a  =  53° 

a  =  53° 

a  =  54° 

a  =  55° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

2738 

5825 

27  2 

59  19 

2625 

60  12 

2548 

61  5 

2511 

6157 

2434 

6249 

2356 

6340 

46 

28  9 

52 

32 

46 

55 

38 

26  17 

31 

39 

6222 

25  1 

63  13 

2422 

64  4 

47 

40 

5920 

28  2 

60  13 

2724 

61  5 

46 

57 

26  7 

48 

28 

38 

48 

29 

48 

2911 

49 

32 

41 

53 

33 

2714 

6224 

34 

63  15 

54 

64  4 

25  14 

54 

49 

41 

6018 

29  1 

61  10 

2821 

62  1 

41 

52 

27  1 

42 

2620 

31 

39 

6520 

50 

3010 

48 

30 

40 

49 

30 

28  8 

6320 

27 

64  9 

46 

58 

26  4 

46 

51 

39 

61  19 

58 

62  10 

29  17 

63  0 

35 

49 

53 

38 

2711 

6526 

28 

6613 

52 

31  8 

51 

3026 

41 

44 

30 

29  1 

64  19 

2819 

65  7 

36 

54 

52 

41 

53 

36 

6223 

53 

63  13 

30  10 

64  1 

27 

49 

44 

36 

28  0 

6623 

27  16 

67  9 

54 

32  3 

56 

3120 

45 

36 

33 

52 

6520 

29  8 

66  6 

24 

52 

39 

38 

55 

30 

6330 

46 

64  18 

31  2 

65  5 

3017 

52 

32 

37 

47 

6722 

28  2 

68  7 

56 

57 

64  5 

32  12 

52 

27 

38 

41 

6624 

56 

67  9 

2910 

53 

24 

37 

57 

3323 

40 

37 

6526 

51 

66  12 

31  5 

57 

30  19 

41 

32 

6825 

45 

69  8 

58 

48 

65  16 

33  2 

66  2 

3215 

47 

28 

6731 

41 

6814 

54 

57 

29  6 

39 

59 

34  13 

53 

26 

38 

39 

6722 

51 

68  5 

31  3 

47 

3015 

6929 

27 

7010 

60 

37 

6631 

50 

6714 

33  2 

58 

32  13 

40 

25 

6921 

36 

70  2 

47 

42 

61 

35  1 

67  9 

3413 

52 

24 

6834 

35 

69  15 

46 

56 

56 

36 

30  7 

71  15 

62 

24 

48 

35 

6830 

45 

69  11 

56 

51 

32  6 

7031 

31  16 

71  10 

26 

48 

63 

46 

6828 

56 

69  9 

34  6 

49 

33  16 

7028 

26 

71  7 

35 

45 

44 

7222 

64 

36  8 

69  8 

35  17 

48 

27 

7027 

36 

71  6 

45 

43 

53 

7220 

31  2 

56 

65 

29 

50 

38 

7028 

47 

71  6 

55 

44 

33  3 

7220 

3211 

56 

19 

7331 

66 

49 

7032 

58 

71  9 

35  6 

46 

3413 

7222 

21 

58 

28 

7332 

36 

74  6 

67 

37  9 

71  14 

36  17 

51 

24 

7226 

31 

73  1 

38 

7336 

45 

74  9 

52 

42 

68 

28 

58 

35 

7233 

42 

73  7 

48 

41 

55 

74  14 

33  1 

46 

32  8 

75  18 

69 

46 

7242 

53 

7316 

59 

49 

35  5 

7421 

34  11 

53 

17 

7524 

23 

55 

70 

38  3 

7327 

37  10 

59 

3615 

7431 

21 

75  2 

26 

7533 

32 

76  3 

37 

7632 

71 

20 

74  12 

26 

7443 

31 

75  14 

36 

44 

41 

76  13 

46 

42 

51 

77  9 

72 

36 

58 

41 

7528 

46 

57 

50 

7626 

55 

54 

59 

7721 

33  4 

47 

73 

51 

7544 

56 

7613 

37  0 

7641 

36  4 

77  8 

35  8 

7735 

3412 

78  1 

16 

7826 

74 

39  6 

7631 

3810 

59 

13 

7725 

17 

51 

21 

7816 

24 

41 

28 

79  5 

75 

19 

77  19 

23 

7745 

26 

78  10 

29 

7834 

33 

58 

35 

7921 

39 

44 

76 

32 

78  7 

35 

7832 

38 

55 

41 

79  18 

44 

7940 

46 

80  2 

49 

8023 

77 

44 

56 

47 

79  19 

49 

7941 

52 

80  2 

54 

8023 

56 

43 

59 

81  3 

78 

55 

7945 

57 

80  6 

59 

8027 

37  2 

46 

36  4 

81  6 

35  6 

81  25 

34  8 

43 

79 

40  5 

8035 

39  7 

54 

38  9 

81  13 

11 

8131 

13 

49 

14 

82  7 

16 

8223 

80 

15 

81  25 

16 

8142 

18 

82  0 

19 

82  16 

21 

8233 

22 

49 

24 

83  4 

81 

23 

82  15 

24 

8231 

26 

47 

27 

83  2 

28 

8317 

29 

8331 

31 

45 

82 

31 

83  6 

32 

8320 

33 

8334 

34 

48 

35 

84  1 

36 

8413 

37 

8426 

83 

38 

57 

38 

84  10 

39 

8422 

40 

8434 

41 

45 

41 

56 

42 

85  7 

84 

44 

8448 

44 

59 

45 

8510 

45 

8520 

46 

8530 

46 

8539 

47 

49 

85 

49 

8540 

49 

8549 

49 

58 

50 

86  6 

50 

8615 

50 

8623 

51 

8630 

86 

53 

8632 

53 

8639 

53 

8646 

53 

53 

54 

87  0 

54 

87  6 

54 

87  12 

87 

56 

8724 

56 

8729 

56 

8734 

56 

8739 

57 

45 

57 

49 

57 

54 

88 

58 

88  16 

58 

88  19 

58 

8823 

58 

8826 

59 

8830 

59 

8833 

59 

8836 

89 

41  0 

89  8 

40  0 

89  10 

39  0 

8911 

38  0 

8913 

37  0 

89  15 

36  0 

89  16 

35  0 

89  18 

90 

-  0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

308 


Table  13.    Kelvin's  Snmner  Line  Table 


b 

a  =  56° 

a  =  57° 

a  =  58° 

a  =  59° 

a  =  60° 

a  =  61° 

a  =  62° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

56  0 

0  0 

57  0 

0  0 

58  0 

0  0 

59  0 

0  0 

60  0 

0  0 

61  0 

0  0 

62  0 

1 

34 

0 

33 

0 

32 

0 

31 

0 

30 

0 

29 

0 

28 

0 

2 

1  7 

1 

1  5 

1 

1  4 

1 

1  2 

1 

1  0 

1 

58 

1 

56 

1 

3 

41 

2 

38 

2 

35 

2 

33 

2 

30 

2 

1  27 

2 

125 

2 

4 

214 

4 

2  11 

4 

2  7 

4 

2  4 

4 

2  0 

4 

56 

4 

53 

3 

5 

48 

6 

43 

6 

39 

6 

34 

6 

30 

6 

225 

6 

221 

5 

6 

321 

9 

316 

9 

311 

8 

3  5 

8 

3  0 

8 

54 

8 

49 

8 

7 

54 

12 

48 

12 

42 

11 

36 

11 

30 

11 

323 

11 

3  17 

11 

8 

428 

16 

421 

15 

4  14 

15 

4  7 

15 

59 

14 

52 

14 

45 

14 

9 

5  1 

20 

53 

19 

45 

19 

37 

19 

429 

18 

421 

18 

4  13 

18 

10 

34 

24 

526 

24 

517 

24 

5  8 

23 

59 

23 

50 

22 

41 

22 

11 

6  7 

29 

58 

29 

48 

29 

38 

28 

529 

28 

519 

27 

5  8 

26 

12 

40 

35 

630 

35 

620 

34 

6  9 

33 

58 

33 

47 

32 

36 

31 

13 

713 

41 

7  2 

41 

51 

40 

39 

39 

628 

38 

616 

38 

6  4 

37 

14 

46 

48 

34 

47 

722 

46 

7  9 

45 

57 

44 

44 

44 

31 

43 

15 

8  19 

55 

8  6 

54 

53 

53 

39 

52 

726 

51 

7  13 

50 

59 

49 

16 

52 

57  3 

38 

58  2 

824 

59  0 

8  9 

59 

55 

58 

41 

57 

726 

56 

17 

925 

11 

9  10 

10 

55 

8 

39 

60  7 

824 

61  6 

8  9 

62  4 

53 

63  3 

18 

57 

19 

42 

18 

926 

17 

9  9 

15 

53 

14 

37 

12 

820 

11 

19 

1029 

28 

1013 

27 

56 

26 

39 

24 

922 

22 

9  5 

20 

47 

19 

20 

11  1 

38 

44 

36 

1027 

35 

10  9 

33 

51 

31 

33 

29 

9  14 

27 

21 

33 

48 

11  15 

46 

57 

45 

38 

43 

10  19 

40 

10  0 

38 

41 

36 

22 

12  5 

59 

46 

57 

1127 

55 

11  7 

53 

48 

50 

28 

48 

10  8 

45 

23 

37 

58  10 

12  17 

59  8 

57 

60  6 

36 

61  3 

11  16 

62  1 

55 

58 

34 

55 

24 

13  9 

22 

48 

19 

1227 

17 

12  5 

14 

44 

12 

11  22 

63  9 

11  0 

64  5 

25 

40 

34 

13  19 

31 

57 

29 

34 

26 

12  12 

23 

49 

20 

26 

16 

26 

14  11 

47 

49 

44 

1326 

41 

13  3 

38 

40 

35 

1216 

31 

52 

27 

27 

42 

59  0 

14  19 

57 

55 

54 

31 

50 

13  7 

47 

43 

43 

12  18 

39 

28 

15  13 

14 

49 

60  10 

1424 

61  7 

59 

62  3 

34 

59 

13  9 

55 

44 

51 

29 

44 

28 

15  19 

24 

53 

21 

1427 

17 

14  1 

63  12 

36 

64  8 

13  9 

65  4 

30 

1614 

43 

48 

39 

1522 

35 

55 

31 

28 

26 

14  2 

21 

34 

17 

31 

44 

58 

1617 

54 

50 

50 

1523 

45 

55 

40 

28 

35 

59 

30 

32 

17  14 

6014 

46 

61  10 

16  18 

62  5 

50 

63  0 

1522 

55 

53 

49 

1424 

44 

33 

44 

30 

17  15 

26 

46 

21 

16  17 

15 

48 

6410 

1519 

65  4 

49 

58 

34 

1813 

47 

44 

42 

17  14 

37 

44 

31 

16  14 

25 

44 

19 

15  13 

6613 

35 

42 

61  5 

1812 

59 

42 

54 

1711 

48 

40 

41 

16  9 

35 

37 

28 

36 

19  11 

23 

40 

62  17 

18  9 

6311 

37 

64  5 

17  6 

58 

34 

51 

16  1 

44 

37 

40 

41 

19  8 

35 

36 

29 

18  3 

22 

31 

65  15 

58 

66  7 

25 

67  0 

38 

20  8 

62  0 

36 

54 

19  3 

47 

29 

40 

56 

32 

1722 

24 

48 

16 

39 

36 

20 

20  3 

63  13 

29 

64  6 

55 

58 

1821 

50 

46 

42 

1711 

33 

40 

21  4 

40 

30 

33 

55 

25 

1920 

65  17 

45 

66  8 

18  10 

67  0 

34 

50 

41 

31 

63  1 

56 

53 

2021 

45 

45 

36 

19  9 

27 

33 

18 

56 

68  8 

42 

58 

23 

21  22 

64  14 

46 

65  5 

2010 

56 

33 

47 

56 

37 

1818 

26 

43 

2225 

45 

48 

36 

21  11 

26 

34 

6617 

56 

67  7 

19  19 

56 

40 

45 

44 

51 

64  7 

22  14 

58 

36 

48 

58 

38 

2019 

27 

41 

6816 

19  2 

69  4 

45 

23  17 

30 

39 

6520 

22  0 

6610 

2122 

59 

42 

48 

20  3 

36 

23 

24 

Table  13.    Kelvin's  Simmer  Line  Table 


309 


b 

a  -  56° 

a  =  57° 

a  =  58° 

a  =  59° 

a  =  60° 

a  =  61° 

a  =  62° 

K 

Q 

K 

3Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

23  17 

6430 

2239 

6520 

22  0 

66  10 

21  22 

6659 

2042 

6748 

20  3 

6836 

1923 

6924 

46 

43 

54 

23  4 

43 

24 

32 

45 

6721 

21  5 

68  9 

25 

56 

44 

44 

47 

24  8 

65  18 

28 

66  7 

48 

55 

22  8 

43 

27 

31 

46 

6917 

20  5 

70  4 

48 

33 

43 

52 

31 

2311 

6718 

30 

68  6 

49 

53 

21  7 

39 

25 

25 

49 

58 

66  8 

24  16 

55 

34 

42 

52 

29 

2210 

69  15 

28 

70  1 

45 

46 

50 

2522 

34 

39 

6720 

57 

68  7 

23  14 

53 

31 

38 

48 

23 

21  5 

71  8 

51 

46 

67  0 

25  2 

46 

24  19 

32 

36 

6917 

52 

70  2 

22  8 

46 

24 

30 

52 

26  9 

27 

25 

68  12 

41 

57 

57 

42 

23  12 

26 

28 

71  9 

43 

52 

53 

32 

54 

47 

39 

25  2 

6923 

24  17 

70  7 

32 

50 

47 

33 

22  1 

72  15 

54 

54 

6822 

26  9 

69  6 

23 

50 

37 

33 

52 

71  15 

23  6 

57 

19 

39 

55 

2716 

51 

30 

34 

44 

7017 

57 

59 

24  11 

41 

24 

7222 

37 

73  2 

56 

37 

6920 

51 

70  2 

26  4 

44 

25  17 

71  26 

29 

72  7 

42 

47 

54 

26 

57 

58 

50 

2711 

31 

2£ 

71  12 

36 

53 

47 

33 

24  0 

73  12 

23  11 

51 

58 

2818 

7020 

31 

71  1 

42 

41 

54 

7220 

25  5 

59 

17 

38 

28 

7416 

59 

38 

51 

50 

31 

27  1 

72  10 

2612 

48 

23 

7326 

33 

74  4 

44 

41 

60 

58 

71  22 

28  9 

72  1 

19 

39 

29 

73  17 

40 

54 

49 

30 

59 

75  7 

61 

2917 

54 

27 

32 

37 

73  9 

46 

46 

56 

7422 

25  5 

57 

2414 

33 

62 

35 

7226 

45 

73  3 

54 

39 

27  3 

74  15 

2612 

50 

21 

7525 

29 

59 

63 

53 

59 

29  2 

35 

2811 

7410 

19 

45 

27 

75  19 

36 

53 

44 

7626 

64 

3010 

7332 

19 

74  7 

27 

41 

35 

75  15 

42 

48 

50 

7621 

58 

53 

65 

27 

74  5 

35 

39 

42 

75  12 

50 

45 

57 

7617 

26  4 

49 

25  11 

7720 

66 

43 

39 

50 

7512 

57 

44 

28  4 

7616 

2711 

47 

17 

77  18 

24 

48 

67 

59 

7514 

30  5 

46 

29  12 

7617 

18 

47 

24 

7717 

30 

47 

36 

78  16 

68 

31  14 

49 

20 

7620 

26 

50 

31 

77  19 

37 

48 

43 

7816 

48 

44 

69 

28 

7625 

34 

54 

39 

7723 

44 

51 

50 

78  19 

55 

46 

26  0 

7913 

70 

42 

77  1 

47 

7729 

52 

56 

57 

7823 

28  2 

50 

27  6 

79  16 

11 

42 

71 

55 

37 

31  0 

78  4 

30  4 

7830 

9  9 

56 

13 

7921 

17 

46 

21 

8011 

72 

32  8 

7814 

12 

39 

16 

79  4 

20 

7929 

24 

53 

27 

8017 

31 

40 

73 

20 

51 

23 

79  15 

27 

39 

30 

80  2 

34 

8025 

37 

48 

41 

81  10 

74 

31 

7928 

34 

51 

37 

8014 

40 

36 

44 

57 

46 

81  19 

50 

40 

75 

42 

80  6 

44 

8027 

47 

49 

50 

81  10 

53 

81  30 

55 

50 

58 

8210 

76 

52 

44 

54 

81  4 

56 

81  24 

59 

44 

29  1 

82  3 

8  3 

8222 

27  6 

40 

77 

33  1 

81  22 

32  3 

41 

31  5 

82  0 

0  7 

82  18 

9 

36 

11 

54 

13 

83  11 

78 

10 

82  1 

11 

82  19 

13 

36 

15 

53 

17 

83  9 

18 

8326 

20 

41 

79 

18 

40 

19 

56 

21 

83  12 

22 

8328 

24 

43 

25 

58 

26 

8412 

80 

25 

319 

26 

8334 

28 

48 

29 

84  3 

30 

8416 

31 

8430 

32 

43 

81 

32 

59 

32 

84  12 

34 

8425 

35 

38 

36 

i  50 

37 

85  3 

37 

85  15 

82 

38 

8438 

38 

50 

39 

85  2 

40 

85  13 

41 

8524 

42 

35 

42 

46 

83 

43 

5  18 

43 

528 

44 

39 

45 

49 

45 

58 

46 

86  8 

46 

86  18 

84 

47 

58 

48 

6  7 

48 

8616 

49 

S624 

49 

8633 

50 

41 

50 

49 

85 

51 

638 

52 

46 

52 

53 

52 

37  0 

52 

7  7 

53 

8714 

53 

8721 

86 

54 

718 

55 

724 

55 

8730 

55 

36 

55 

42 

55 

47 

56 

52 

87 

57 

59 

57 

8  3 

57 

88  8 

57 

38  12 

57 

816 

57 

8820 

58 

8824 

88 

59 

839 

59 

42 

59 

45 

59 

48 

59 

51 

59 

53 

59 

56 

89 

4  0 

9  19 

3  0 

921 

2  0 

923 

1  0 

3924 

0  0 

925 

9  0 

3927 

8  0 

8928 

90 

0 

0  0 

0 

0  0 

0 

0  0 

0 

0  0 

0 

0  0 

0 

30  0 

0 

90  0 

310 


Table  13.    Kelvin's  Sumner  Line  Table 


a  =  63° 

a  =  64° 

a  =  65° 

a  =  66° 

a  =  67° 

a  =  68° 

a  =  69° 

b 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

63  0 

0  0 

64  0 

0  0 

65  0 

0  0 

66  0 

0  0 

67  0 

0  0 

68  0 

0  0 

69  0 

1 

27 

0 

26 

0 

25 

0 

24 

0 

23 

0 

22 

0 

22 

0 

f\ 

54 

1 

53 

1 

51 

1 

49 

1 

47 

] 

45 

] 

43 

1 

o 

1  22 

2 

1  19 

o 

1  16 

r 

A 

1  13 

c 
£ 

1  10 

f 

1  7 

f 

1  5 

2 

4 

49 

3 

45 

3 

41 

3 

38 

f 
«_ 

34 

i 

30 

t 

26 

3 

5 

6 

2  16 
43 

7 

2  11 
38 

7 

2  7 
32 

7 

2  2 

26 

7 

57 

220 

7 

52 
2  15 

7 

47 
2  9 

4 
6 

7 

3  10 

10 

3  4 

10 

57 

10 

50 

9 

44 

9 

37 

9 

30 

8 

8 

37 

13 

30 

13 

322 

13 

315 

12 

3  7 

12 

59 

12 

52 

11 

9 

4  4 

17 

56 

17 

47 

16 

39 

16 

30 

15 

322 

15 

313 

14 

10 

31 

21 

422 

21 

4  12 

20 

4  3 

20 

53 

19 

44 

18 

34 

17 

11 

58 

26 

48 

25 

37 

24 

27 

24 

4  16 

23 

4  6 

22 

55 

21 

12 

525 

31 

5  14 

30 

5  2 

29 

51 

28 

39 

27 

28 

26 

4  16 

25 

13 

52 

36 

40 

35 

27 

34 

5  15 

33 

5  2 

32 

50 

31 

37 

29 

14 

6  18 

42 

6  5 

40 

52 

39 

39 

38 

25 

37 

5  12 

36 

58 

34 

15 

45 

48 

31 

46 

617 

45 

6  3 

44 

48 

42 

34 

41 

519 

39 

16 

711 

54 

56 

53 

41 

51 

26 

50 

611 

48 

56 

47 

40 

45 

17 

38 

64  1 

722 

65  0 

7  6 

58 

50 

56 

34 

54 

617 

53 

6  1 

51 

18 

8  4 

9 

47 

7 

30 

66  5 

7  13 

67  3 

56 

68  1 

39 

59 

22 

57 

19 

30 

17 

812 

15 

54 

12 

37 

10 

7  19 

8 

7  0 

69  6 

42 

70  3 

20 

56 

25 

37 

23 

818 

20 

8  0 

18 

41 

15 

22 

13 

7  3 

10 

21 

922 

34 

9  2 

31 

42 

28 

23 

26 

8  3 

23 

43 

20 

23 

17 

22 

48 

43 

27 

40 

9  6 

37 

46 

34 

25 

31 

8  4 

28 

43 

24 

23 

10  13 

52 

52 

49 

30 

46 

9  9 

43 

47 

39 

25 

36 

8  3 

32 

24 

39 

65  2 

10  16 

59 

54 

56 

31 

52 

9  9 

48 

46 

44 

23 

40 

25 

11  4 

13 

41 

66  9 

1017 

67  6 

54 

68  2 

30 

57 

9  7 

53 

43 

49 

26 

29 

24 

11  5 

20 

41 

16 

10  16 

12 

52 

69  7 

27 

70  2 

9  2 

58 

27 

54 

35 

29 

31 

11  4 

26 

38 

22 

1013 

17 

48 

12 

22 

71  7 

28 

12  19 

47 

53 

42 

27 

37 

11  0 

32 

34 

27 

10  8 

22 

41 

17 

29 

43 

59 

12  16 

54 

50 

49 

22 

43 

55 

38 

28 

32 

10  0 

27 

30 

13  7 

66  11 

40 

67  6 

12  12 

68  1 

44 

55 

11  16 

49 

48 

43 

19 

37 

31 

31 

24 

13  3 

19 

34 

13 

12  6 

69  7 

37 

70  0 

11  8 

54 

38 

47 

32 

55 

38 

26 

32 

56 

25 

27 

19 

57 

12 

27 

71  5 

57 

58 

33 

14  19 

52 

49 

45 

1318 

38 

48 

31 

12  17 

24 

46 

17 

11  15 

72  9 

34 

43 

67  6 

1412 

59 

40 

52 

13  9 

44 

37 

37 

12  5 

29 

33 

21 

35 

15  6 

21 

34 

68  13 

14  2 

69  6 

30 

58 

57 

50 

24 

41 

51 

33 

36 

29 

36 

56 

28 

23 

20 

50 

70  12 

13  17 

71  3 

43 

54 

12  9 

45 

37 

52 

51 

1518 

43 

44 

34 

1410 

26 

36 

16 

13  2 

72  7 

27 

57 

38 

16  14 

68  7 

40 

59 

15  5 

49 

30 

40 

55 

30 

20 

20 

45 

73  10 

39 

36 

24 

16  1 

69  15 

26 

70  5 

50 

55 

14  14 

44 

38 

34 

13  2 

23 

40 

58 

41 

22 

31 

46 

21 

15  9 

71  10 

33 

59 

56 

48 

19 

37 

41 

1720 

58 

43 

48 

16  6 

37 

28 

26 

51 

7214 

14  14 

73  2 

36 

51 

42 

41 

6916 

17  4 

70  5 

26 

53 

47 

42 

15  9 

29 

31 

17 

53 

74  5 

43 

18  2 

34 

24 

22 

45 

71  10 

16  6 

58 

27 

45 

48 

32 

14  9 

19 

44 

23 

52 

44 

40 

17  4 

27 

25 

72  14 

45 

73  1 

15  5 

48 

25 

34 

45 

44 

70  11 

18  4 

58 

23 

45 

43 

31 

6  2 

18 

22 

74  3 

41 

49 

Table  13.    Kelvin's  Sumner  Line  Table 


311 


b 

a  -  63° 

a  =  64° 

a  =  65° 

a  =  66° 

a  =  67° 

a  =  68° 

a  =  69° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

1844 

70  11 

8  4 

7058 

1723 

7145 

1643 

7231 

16  2 

73  18 

1522 

74  3 

1441 

7449 

46 

19  4 

30 

23 

71  17 

42 

72  3 

17  1 

49 

19 

34 

38 

19 

56 

75  4 

47 

24 

50 

42 

36 

18  0 

21 

18 

73  7 

36 

51 

54 

36 

15  11 

20 

48 

43 

71  10 

9  1 

55 

18 

40 

35 

25 

53 

74  8 

16  10 

52 

26 

36 

49 

20  2 

31 

19 

72  15 

36 

59 

52 

43 

17  9 

26 

26 

75  9 

41 

52 

50 

21 

52 

37 

36 

53 

73  19 

18  9 

74  2 

25 

44 

41 

26 

56 

76  8 

51 

40 

72  13 

55 

56 

19  10 

39 

25 

21 

41 

75  3 

56 

44 

1610 

25 

52 

58 

35 

20  12 

73  17 

27 

59 

41 

40 

56 

21 

17  10 

76  2 

24 

42 

53 

21  16 

57 

29 

38 

43 

74  19 

57 

75  0 

1811 

40 

24 

20 

38 

59 

54 

33 

7320 

46 

74  0 

59 

40 

1913 

20 

26 

59 

38 

38 

51 

7717 

55 

50 

43 

21  3 

22 

2015 

75  1 

28 

40 

40 

76  19 

52 

57 

17  4 

35 

56 

22  7 

74  6 

19 

45 

30 

23 

43 

76  1 

54 

39 

18  6 

77  16 

17 

53 

57 

23 

30 

34 

75  7 

45 

45 

57 

22 

19  8 

59 

19 

35 

29 

7811 

58 

39 

54 

49 

30 

21  0 

76  7 

2011 

43 

21 

7719 

32 

55 

41 

30 

59 

54 

75  18 

22  4 

54 

14 

30 

24 

77  5 

34 

40 

44 

7815 

53 

49 

60 

23  9 

42 

19 

7618 

28 

53 

37 

27 

47 

78  1 

56 

35 

18  5 

79  8 

61 

24 

76  7 

33 

42 

42 

7716 

50 

49 

59 

22 

19  8 

55 

16 

27 

62 

38 

33 

46 

77  6 

55 

39 

21  3 

7811 

20  11 

44 

19 

7916 

27 

47 

63 

52 

59 

59 

31 

22  7 

78  3 

15 

34 

23 

79  6 

30 

37 

37 

80  7 

64 

24  5 

7725 

23  12 

56 

19 

27 

27 

57 

34 

28 

41 

58 

47 

27 

60 

18 

51 

24 

7821 

31 

51 

38 

7920 

45 

50 

51 

8019 

57 

47 

66 

30 

78  18 

36 

47 

43 

79  16 

49 

44 

55 

8012 

20  1 

40 

19  7 

81  8 

67 

42 

45 

48 

79  13 

54 

41 

59 

80  8 

21  5 

35 

10 

81  2 

16 

28 

68 

54 

7912 

59 

39 

23  4 

80  6 

22  9 

32 

15 

58 

19 

24 

25 

49 

69 

25  5 

39 

2410 

80  5 

14 

31 

19 

56 

24 

8121 

28 

46 

33 

82  10 

70 

15 

80  7 

20 

32 

24 

56 

28 

81  20 

33 

44 

37 

82  8 

41 

31 

71 

25 

35 

29 

59 

33 

8122 

37 

45 

41 

82  8 

45 

30 

49 

53 

72 

35 

81  3 

38 

81  26 

42 

48 

45 

82  10 

49 

32 

52 

53 

56 

83  14 

73 

44 

32 

47 

53 

50 

8214 

53 

35 

57 

56 

59 

83  16 

20  3 

36 

74 

53 

82  0 

55 

8220 

58 

40 

23  1 

83  0 

22  4 

8320 

21  6 

39 

9 

58 

75 

26  1 

29 

25  3 

48 

24  6 

83  7 

8 

26 

11 

44 

13 

84  2 

15 

8420 

76 

8 

58 

10 

83  16 

13 

34 

15 

51 

17 

84  8 

19 

25 

21 

42 

77 

15 

8328 

17 

44 

19 

84  1 

21 

8417 

23 

33 

25 

48 

26 

85  4 

78 

22 

57 

23 

84  13 

25 

28 

27 

43 

28 

57 

30 

8512 

31 

26 

79 

28 

8427 

29 

41 

31 

55 

32 

85  9 

33 

8522 

35 

36 

36 

49 

80 

33 

57 

34 

85  9 

36 

8522 

37 

35 

38 

47 

39 

59 

40 

86  11 

81 

38 

8527 

39 

38 

40 

50 

41 

86  1 

42 

86  12 

43 

8623 

44 

34 

82 

43 

57 

43 

86  7 

44 

86  17 

45 

27 

46 

37 

47 

47 

47 

57 

83 

47 

8627 

47 

36 

48 

45 

49 

54 

49 

87  2 

50 

87  11 

50 

87  19 

84 

50 

57 

51 

87  5 

51 

87  12 

52 

8720 

52 

28 

53 

35 

53 

42 

85 

53 

8727 

54 

34 

54 

40 

54 

47 

54 

53 

55 

59 

55 

88  5 

86 

56 

58 

56 

88  3 

56 

88  8 

56 

8813 

56 

8818 

57 

8823 

57 

28 

87 

58 

8828 

58 

32 

58 

36 

58 

40 

58 

44 

58 

47 

58 

51 

88 

59 

59 

59 

89  1 

59 

89  4 

59 

89  7 

59 

89  9 

59 

89  12 

59 

89  14 

89 

27  0 

8929 

26  0 

31 

25  0 

32 

24  0 

33 

23  0 

34 

22  0 

36 

21  0 

37 

90 

0 

90  0 

0 

90  0 

0 

900 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

312 


Table  13.    Kelvin's  Sumner  Line  Table 


h 

a  =  70° 

a  =  71° 

a  =  72° 

a  =  73° 

a  =  74° 

a  =  75° 

a  =  76° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

70  0 

0  0 

71  0 

0  0 

72  0 

0  0 

73  0 

0  0 

74  0 

0  0 

75  0 

0  0 

76  0 

1 

21 

0 

20 

0 

19 

0 

18 

0 

17 

0 

16 

0 

15 

0 

2 

41 

1 

39 

1 

37 

1 

35 

1 

33 

1 

31 

1 

29 

0 

3 

1  2 

2 

59 

2 

56 

1 

53 

1 

50 

1 

47 

1 

44 

1 

4 

22 

3 

1  18 

3 

1  14 

2 

1  10 

2 

1  6 

2 

1  2 

2 

58 

2 

5 

43 

4 

38 

4 

33 

4 

28 

4 

23 

3 

18 

3 

1  13 

3 

6 

2  3 

6 

57 

6 

51 

6 

45 

5 

39 

5 

33 

5 

27 

4 

7 

23 

8 

217 

8 

2  9 

8 

2  3 

7 

56 

7 

49 

6 

41 

6 

8 

44 

11 

36 

10 

28 

10 

20 

9 

2  12 

9 

2  4 

8 

56 

8 

9 

3  4 

14 

55 

13 

46 

13 

37 

12 

28 

11 

19 

10 

2  10 

10 

10 

24 

17 

3  15 

16 

3  5 

16 

55 

15 

45 

14 

35 

13 

25 

12 

11 

45 

20 

34 

19 

23 

19 

3  12 

18 

3  1 

17 

50 

16 

39 

15 

12 

4  5 

24 

53 

23 

41 

22 

29 

21 

17 

20 

3  5 

19 

53 

18 

13 

25 

28 

412 

27 

59 

26 

46 

25 

33 

23 

20 

22 

3  7 

21 

14 

45 

33 

31 

31 

417 

30 

4  3 

29 

49 

27 

35 

25 

21 

24 

15 

5  5 

38 

50 

36 

35 

34 

20 

33 

4  5 

31 

50 

29 

35 

27 

16 

25 

43 

5  9 

41 

53 

39 

37 

37 

21 

35 

4  5 

33 

49 

31 

17 

44 

49 

28 

46 

511 

44 

54 

42 

37 

40 

20 

38 

4  3 

35 

18 

6  4 

55 

46 

52 

29 

49 

511 

47 

53 

4  5 

35 

42 

17 

40 

19 

24 

71  1 

6  5 

58 

46 

55 

28 

52 

5  9 

50 

50 

47 

31 

44 

20 

43 

7 

24 

72  4 

6  4 

73  1 

44 

58 

25 

55 

5  5 

52 

45 

49 

21 

7  3 

14 

42 

11 

21 

7 

6  1 

74  4 

40 

75  1 

19 

57 

59 

54 

22 

22 

21 

7  0 

18 

39 

14 

17 

10 

56 

7 

34 

76  3 

5  12 

59 

23 

41 

29 

18 

25 

56 

21 

34 

17 

611 

13 

48 

9 

26 

77  4 

24 

8  0 

37 

36 

32 

7  13 

28 

50 

24 

26 

19 

6  3 

15 

39 

10 

25 

19 

45 

54 

40 

30 

35 

7  6 

31 

41 

26 

17 

21 

52 

16 

26 

38 

53 

8  12 

48 

47 

43 

22 

38 

56 

33 

31 

27 

6  5 

22 

27 

56 

72  2 

30 

56 

8  4 

51 

38 

46 

7  11 

40 

45 

34 

18 

28 

28 

9  15 

11 

48 

73  5 

21 

59 

54 

54 

26 

48 

59 

41 

31 

35 

29 

33 

20 

9  5 

14 

37 

74  8 

8  9 

75  2 

41 

55 

713 

48 

44 

42 

30 

51 

30 

22 

24 

53 

17 

25 

10 

55 

76  3 

26 

56 

57 

49 

31 

10  9 

40 

39 

34 

9  9 

26 

40 

19 

810 

11 

40 

77  4 

7  10 

56 

32 

27 

51 

56 

44 

25 

36 

55 

28 

24 

20 

53 

12 

22 

78  4 

33 

44 

73  2 

1013 

54 

41 

46 

9  10 

37 

38 

29 

8  6 

20 

34 

11 

34 

11  2 

13 

30 

74  4 

57 

56 

25 

46 

52 

38 

19 

28 

46 

19 

35 

19 

24 

46 

15 

1013 

75  6 

39 

56 

9  6 

47 

32 

37 

58 

27 

36 

36 

36 

11  2 

26 

28 

16 

54 

76  6 

19 

56 

45 

46 

810 

36 

37 

53 

48 

18 

37 

43 

27 

10  8 

17 

33 

77  6 

58 

55 

22 

44 

38 

12  9 

74  0 

34 

49 

58 

38 

22 

27 

46 

16 

9  10 

78  4 

34 

53 

39 

26 

12 

50 

75  1 

11  13 

50 

36 

38 

59 

26 

22 

14 

46 

79  2 

40 

42 

25 

12  5 

13 

28 

76  1 

50 

49 

1012 

37 

34 

24 

57 

11 

41 

58 

38 

20 

26 

42 

13 

11  4 

77  0 

25 

47 

46 

34 

9  8 

21 

42 

13  14 

52 

35 

39 

56 

26 

17 

12 

38 

58 

58 

44 

19 

30 

43 

29 

75  6 

50 

52 

12  10 

38 

30 

24 

50 

78  9 

10  10 

55 

30 

40 

44 

44 

20 

13  4 

76  5 

24 

51 

43 

36 

11  2 

21 

22 

79  5 

41 

50 

45 

59 

34 

18 

19 

38 

77  4 

56 

48 

14 

32 

33 

16 

51 

80  0 

Table  13.    Kelvin's  Sumner  Line  Table 


313 


b 

a  =  70° 

a  =  71° 

a  =  73° 

a  =  73° 

a  =  74° 

a  =  75° 

a  =  76° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

359 

534 

3  18 

6  19 

238 

7  4 

156 

748 

1  14 

7832 

033 

79  16 

951 

80  0 

46 

4  14 

49 

32 

33 

51 

17 

2  9 

8  1 

26 

44 

44 

27 

0  1 

10 

47 

29 

6  4 

46 

47 

3  4 

30 

21 

13 

38 

56 

55 

39 

11 

21 

48 

43 

19 

4  0 

7  1 

17 

44 

33 

26 

49 

79  8 

11  6 

50 

21 

32 

49 

57 

34 

13 

16 

29 

58 

45 

39 

12  0 

21 

16 

80  2 

31 

43 

50 

5  11 

50 

26 

31 

41 

812 

57 

53 

11 

33 

26 

14 

41 

54 

51 

25 

7  6 

39 

46 

53 

26 

3  8 

79  7 

22 

46 

36 

26 

50 

81  5 

52 

38 

22 

52 

8  2 

4  5 

41 

19 

21 

33 

59 

46 

38 

59 

16 

53 

51 

39 

5  4 

18 

17 

56 

30 

35 

43 

8013 

56 

50 

11  8 

28 

54 

6  4 

55 

16 

34 

29 

7911 

41 

49 

53 

26 

12  5 

81  3 

17 

40 

55 

16 

8  12 

28 

50 

40 

26 

52 

80  3 

13  3 

40 

14 

16 

26 

52 

56 

28 

30 

40 

9  6 

51 

42 

14  2 

18 

13 

54 

23 

29 

34 

82  4 

57 

40 

47 

51 

23 

5  1 

58 

12 

33 

22 

81  8 

32 

42 

42 

16 

58 

52 

9  5 

6  2 

40 

11 

8014 

22 

48 

31 

22 

41 

55 

50 

28 

59 

7  3 

23 

12 

57 

21 

30 

31 

81  3 

40 

36 

49 

82  9 

58 

41 

60 

14 

41 

22 

80  14 

31 

46 

40 

18 

49 

'  50 

57 

22 

12  6 

54 

61 

24 

80  0 

32 

31 

41 

81  3 

49 

34 

57 

82  5 

13  5 

36 

13 

83  7 

•62 

34 

18 

42 

49 

50 

20 

58 

50 

14  5 

20 

13 

50 

20 

20 

63 

44 

37 

52 

81  7 

59 

37 

15  6 

82  6 

13 

35 

20 

83  4 

27 

33 

64 

54 

56 

7  1 

25 

16  8 

54 

14 

22 

21 

50 

27 

18 

34 

46 

65 

8  3 

81  15 

10 

43 

16 

82  11 

22 

38 

28 

83  5 

34 

32 

40 

59 

66 

12 

35 

18 

82  2 

24 

28 

30 

55 

35 

21 

41 

47 

46 

84  12 

67 

21 

54 

26 

20 

32 

46 

37 

83  11 

42 

36 

47 

84  1 

52 

26 

68 

29 

82  14 

34 

39 

39 

83  4 

44 

28 

49 

52 

53 

16 

58 

40 

69 

37 

34 

42 

58 

46 

22 

51 

45 

55 

84  8 

59 

31 

13  3 

54 

70 

45 

54 

49 

8317 

53 

40 

57 

84  2 

15  1 

24 

14  5 

46 

8 

85  8 

71 

52 

83  15 

56 

36 

59 

58 

16  3 

19 

7 

40 

10 

85  1 

13 

22 

72 

59 

35 

18  2 

56 

17  5 

84  16 

9 

36 

12 

56 

15 

16 

18 

36 

73 

19  6 

56 

8 

84  15 

11 

34 

14 

53 

17 

85  12 

20 

31 

23 

50 

74 

12 

84  16 

14 

35 

17 

53 

19 

8511 

22 

29 

25 

47 

27 

86  4 

75 

18 

37 

20 

54 

22 

8512 

24 

28 

27 

45 

29 

86  2 

31 

18 

76 

23 

58 

25 

85  14 

27 

30 

29 

46 

31 

86  2 

33 

17 

35 

33 

77 

28 

85  19 

30 

34 

31 

49 

33 

86  4 

35 

19 

37 

33 

38 

47 

78 

33 

40 

34 

54 

35 

86  8 

37 

22 

39 

35 

40 

49 

41 

87  2 

79 

37 

86  2 

38 

8614 

39 

27 

41 

40 

42 

52 

43 

87  4 

44 

17 

80 

41 

23 

42 

35 

43 

46 

44 

58 

45 

87  9 

46 

20 

47 

31 

81 

45 

44 

45 

55 

46 

87  5 

47 

8716 

48 

26 

49 

36 

50 

46 

82 

48 

87  6 

48 

87  15 

49 

25 

50 

34 

51 

43 

51 

52 

52 

88  1 

83 

51 

28 

51 

36 

52 

44 

52 

52 

53 

88  0 

53 

88  8 

54 

15 

84 

53 

49 

53 

56 

54 

88  3 

54 

8810 

55 

17 

55 

24 

56 

30 

8 

55 

8811 

55 

8817 

56 

23 

56 

28 

56 

34 

57 

40 

57 

45 

8 

57 

33 

57 

37 

57 

42 

57 

47 

57 

51 

58 

56 

58 

89  0 

8 

58 

55 

58 

58 

58 

89  1 

58 

89  5 

58 

89  8 

59 

89  12 

59 

15 

8 

59 

89  16 

59 

8919 

59 

21 

59 

23 

59 

26 

15  0 

28 

140 

30 

8 

20  0 

38 

19  0 

39 

18  0 

40 

17  0 

42 

16  0 

43 

0 

44 

0 

45 

9 

0 

90  0 

I 

90  0 

I 

90  0 

0 

90  C 

0 

90  0 

0 

90  0 

0 

90  0 

314 


Table  13.    Kelvin's  Sunmer  Line  Table 


a  = 

77° 

a  = 

=  78° 

a  = 

=  79° 

a  = 

=  80° 

a  = 

=  81° 

a  = 

=  83° 

a  - 

=  83° 

1) 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

( 

0  0 

77  0 

0  0 

78  0 

0  0 

79  0 

0  0 

80  0 

0  0 

81  0 

0  0 

82  0 

0  ( 

83  0 

14 

I 

12 

i 

!   11 

0 

10 

0 

I 

0 

8 

0 

4 

0 

1 

27 

I 

25 

i 

23 

0 

21 

o 

19 

0 

17 

0 

15 

0 

J 

41 

; 

37 

34 

; 

31 

; 

28 

• 

25 

] 

22 

1 

< 

54 

2 

50 

2 

46 

i 

42 

1 

38 

1 

33 

1 

29 

1 

5 

6 

1  V 
21 

1 

1  2 
15 

; 

57 
1  9 

, 

52 
1  2 

j 

47 
56 

2 

42 
50 

2 

37 

44 

2 
2 

Q 

34 

A  O 

( 

27 

on 

5 

20 

31 

i 

13 

2O 

4 

1  6 

< 

58 

< 

51 

pro 

3 

o 

9 

4o 

2  1 

9 

Oi 

52 

c 

i 
43 

( 

8 

O 

33 

7 

24 

6 

15 

6 

Oo 

1  6 

• 

5 

10 

14 

11 

2  4 

11 

54 

10 

44 

9 

33 

8 

23 

7 

13 

6 

1] 

28 

\i 

2  5 

54 

K 

31 

c 

2C 

g 

12 

41 

16 

29 

15 

16 

14 

2  i 

13 

52 

12 

40 

10 

27 

9 

13 

54 

19 

41 

18 

28 

16 

14 

15 

2  1 

14 

48 

12 

34 

11 

14 

3  7 

22 

53 

21 

39 

19 

24 

17 

10 

16 

56 

14 

41 

12 

15 

20 

26 

3  5 

24 

50 

22 

34 

20 

19 

18 

2  4 

16 

48 

14 

16 

33 

29 

17 

27 

3  1 

25 

44 

23 

28 

20 

12 

18 

55 

16 

17 

46 

33 

29 

30 

12 

28 

54 

26 

37 

23 

20 

21 

2  2 

18' 

18 

59 

37 

41 

34 

23 

31 

3  4 

29 

46 

26 

28 

23 

9 

20 

19 

412 

41 

53 

38 

34 

35 

14 

32 

55 

29 

36 

26 

16 

23 

20 

25 

45 

4  5 

42 

45 

39 

24 

35 

3  4 

32 

44 

29 

23 

25 

21 

37 

50 

17 

46 

55 

43 

34 

39 

13 

35 

52 

32 

30 

28 

22 

50 

55 

28 

51 

4  6 

47 

44 

43 

22 

39 

59 

35 

37 

30 

23 

5  3 

78  0 

40 

56 

17 

51 

53 

47 

30 

42 

3  7 

38 

44 

33 

24 

15 

C 

Q 

51 

79  1 

27 

56 

4  3 

51 

39 

46 

15 

41 

50 

36 

25 

27 

11 

5  3 

6 

38 

80  1 

13 

55 

47 

50 

22 

44 

57 

39 

26 

39 

17 

14 

11 

48 

6 

22 

81  0 

56 

54 

30 

48 

3  4 

42 

27 

51 

23 

25 

16 

58 

11 

31 

4 

4  5 

58 

37 

52 

10 

45 

28 

6  3 

29 

36 

22 

5  8 

16 

41 

9 

13 

82  2 

45 

56 

17 

49 

29 

15 

35 

47 

28 

18 

21 

50 

14 

21 

7 

52 

83  0 

23 

52 

30 

27 

41 

58 

34 

28 

27 

59 

19 

29 

11 

59 

4 

30 

56 

31 

39 

48 

6  9 

40 

38 

33 

5  8 

24 

37 

16 

4  7 

8 

36 

84  0 

32 

51 

55 

20 

47 

48 

39 

17 

30 

45 

21 

14 

12 

42 

3 

33 

7  2 

79  3 

30 

54 

58 

45 

26 

35 

53 

26 

21 

17 

48 

7 

34 

14 

10 

41 

80  1 

6  8 

51 

34 

41 

5  1 

31 

28 

21 

54 

11 

35 

25 

17 

51 

8 

17 

57 

43 

47 

9 

36 

35 

26 

4  0 

15 

36 

36 

25 

7  1 

15 

27 

81  4 

52 

53 

17 

42 

42 

31 

6 

20 

37 

47 

33 

11 

22 

36 

11 

6  0 

59 

24 

47 

48 

36 

12 

24 

38 

58 

41 

21 

29 

45 

18 

8 

82  5 

32 

53 

55 

41 

18 

28 

39 

8  8 

50 

31 

37 

54 

25 

16 

12 

39 

59 

5  2 

46 

24 

33 

40 

19 

58 

41 

45 

7  3 

32 

24 

18 

46 

83  5 

8 

51 

30 

38 

41 

29 

80  7 

51 

53 

12 

39 

32 

25 

53 

11 

14 

57 

35 

42 

42 

39 

16 

8  0 

31  1 

20 

47 

40 

32 

6  0 

17 

21 

84  2 

41 

47 

43 

49 

25 

9 

10 

29 

55 

48 

39 

7 

23 

27 

8 

46 

52 

44 

59 

34 

18 

18 

37 

S2  2 

56 

46 

14 

30 

33 

14 

52 

57 

45 

9  9 

43 

27 

27 

45 

10 

7  3 

54 

21 

37 

39 

20 

57 

85  2 

Table  13.    Kelvin's  Sumner  Line  Table 


315 


b 

a  =  77° 

a  =  78° 

a  =  79° 

a  =  80° 

a  =  81° 

a  =  82° 

a  =  83° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

9  9 

8043 

827 

8127 

745 

82  10 

7  3 

8254 

621 

8337 

539 

8420 

457 

85  2 

46 

19 

53 

36 

36 

53 

19 

11 

83  1 

28 

43 

45 

26 

5  2 

7 

47 

28 

81  3 

45 

45 

8  1 

27 

18 

9 

34 

50 

51 

32 

7 

13 

48 

37 

13 

53 

54 

9 

35 

25 

16 

41 

57 

56 

38 

12 

18 

49 

46 

23 

9  2 

82  4 

17 

44 

32 

24 

47 

84  4 

6  2 

44 

17 

24 

50 

55 

33 

10 

13 

24 

53 

39 

32 

53 

11 

7 

50 

21 

29 

51 

10  4 

44 

18 

23 

32 

83  2 

45 

40 

59 

18 

13 

57 

26 

35 

52 

13 

55 

26 

33 

39 

11 

52 

48 

7  5 

26 

18 

85  3 

31 

41 

53 

21 

82  6 

34 

43 

46 

20 

58 

57 

11 

33 

23 

10 

35 

46 

54 

29 

17 

41 

53 

53 

29 

8  4 

84  5 

16 

41 

28 

17 

40 

52 

55 

37 

28 

48 

83  3 

9  0 

38 

10 

13 

22 

49 

33 

23 

44 

58 

56 

45 

39 

55 

13 

6 

48 

16 

22 

27 

56 

38 

30 

48 

86  4 

57 

53 

50 

10  2 

24 

13 

57 

22 

31 

32 

85  4 

42 

37 

52 

10 

58 

11  0 

83  1 

9 

34 

19 

84  7 

28 

40 

37 

12 

47 

44 

56 

17 

59 

7 

13 

16 

45 

25 

17 

34 

49 

42 

20 

51 

52 

6  0 

23 

60 

14 

25 

23 

56 

31 

27 

39 

58 

47 

28 

55 

59 

4 

29 

61 

21 

37 

29 

84  7 

37 

37 

44 

85  7 

52 

37 

59 

86  6 

7 

36 

62 

28 

49 

35 

18 

42 

47 

49 

16 

57 

45 

7  3 

13 

11 

42 

63 

34 

84  1 

41 

29 

47 

57 

54 

25 

8  1 

53 

7 

21 

14 

49 

64 

40 

13 

46 

41 

52 

85  8 

59 

35 

5 

86  2 

11 

28 

17 

55 

65 

46 

25 

52 

52 

57 

18 

9  4 

44 

9 

10 

15 

36 

20 

87  2 

66 

52 

38 

57 

85  4 

10  2 

29 

8 

54 

13 

19 

19 

44 

23 

8 

67 

57 

51 

11  2 

15 

7 

39 

12 

86  4 

17 

28 

22 

51 

26 

15 

68 

12  2 

85  3 

7 

27 

11 

50 

16 

13 

21 

36 

25 

59 

29 

22 

69 

7 

16 

12 

39 

15 

86  1 

20 

23 

24 

45 

28 

87  7 

32 

29 

70 

12 

29 

16 

51 

19 

12 

24 

33 

27 

54 

31 

15 

35 

36 

71 

17 

42 

20 

86  3 

23 

23 

27 

43 

30 

87  3 

34 

23 

37 

43 

72 

21 

55 

24 

15 

27 

34 

30 

53 

33 

12 

37 

31 

39 

50 

73 

25 

86  8 

28 

27 

31 

45 

33 

87  3 

36 

21 

39 

39 

i  42 

57 

74 

29 

22 

32 

39 

34 

56 

36 

13 

39 

30 

41 

47 

44 

88  4 

75 

33 

35 

35 

51 

37 

87  7 

39 

23 

42 

39 

43 

55 

46 

11 

76 

37 

48 

38 

87  3 

40 

18 

42 

33 

44 

48 

45 

88  3 

48 

18 

77 

40 

87  2 

41 

16 

43 

30 

45 

44 

46 

58 

47 

11 

49 

25 

78 

43 

15 

44 

28 

46 

41 

47 

54 

48 

88  7 

49 

20 

51 

32 

79 

46 

29 

47 

41 

48 

53 

49 

88  4 

50 

16 

51 

28 

52 

39 

80 

48 

42 

49 

53 

50 

88  4 

51 

15 

52 

25 

53 

36 

54 

47 

81 

50 

56 

51 

88  6 

52 

15 

53 

25 

54 

35 

54 

44 

55 

54 

82 

52 

8810 

53 

18 

54 

27 

54 

36 

55 

44 

55 

53 

56 

89  1 

83 

54 

23 

55 

31 

55 

39 

55 

46 

56 

54 

56 

89  1 

57 

9 

84 

56 

37 

56 

44 

56 

50 

56 

57 

57 

89  3 

57 

10 

58 

16 

85 

57 

51 

57 

56 

57 

89  2 

57 

89  7 

58 

13 

58 

18 

58 

23 

86 

58 

89  5 

58 

89  9 

58 

13 

58 

18 

59 

22 

59 

26 

59 

31 

87 

59 

18 

59 

22 

59 

25 

59 

28 

59 

31 

59 

35 

59 

38 

88 

13  0 

32 

12  0 

34 

11  0 

37 

10  0 

39 

9  0 

41 

8  0 

43 

7  0 

45 

89 

0 

46 

0 

47 

0 

48 

0 

49 

0 

50 

0 

52 

0 

53 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

316 


Table  13.    Kelvin's  Simmer  Line  Table 


1) 

a  = 

84° 

a  = 

85° 

a  = 

86° 

a  = 

87° 

a  = 

88° 

a  = 

89° 

a  = 

90° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

0  0 

84  0 

0  0 

85  0 

0  0 

86  0 

0  0 

87  0 

0  0 

88  0 

0  0 

89  0 

0  0 

90  0 

1 

6 

0 

5 

0 

4 

0 

3 

0 

2 

0 

1 

0 

0 

0 

2 

13 

0 

10 

0 

8 

0 

6 

0 

4 

0 

2 

0 

0 

0 

3 

19 

0 

16 

0 

13 

0 

9 

0 

6 

0 

3 

0 

0 

0 

4 

25 

1 

21 

1 

17 

1 

13 

0 

8 

0 

4 

0 

0 

0 

5 

31 

1 

26 

1 

21 

1 

16 

1 

10 

0 

5 

0 

0 

0 

6 

38 

2 

31 

2 

25 

1 

19 

1 

13 

1 

6 

0 

0 

0 

7 

44 

3 

37 

2 

29 

2 

22 

1 

15 

1 

7 

0 

0 

0 

8 

50 

3 

42 

3 

33 

2 

25 

2 

17 

1 

8 

1 

0 

0 

9 

56 

4 

47 

4 

38 

3 

28 

2 

19 

1 

9 

1 

0 

0 

10 

1  2 

5 

52 

5 

42 

4 

31 

3 

21 

2 

10 

1 

0 

0 

11 

9 

7 

57 

6 

46 

4 

34 

3 

23 

2 

11 

1 

0 

0 

12 

15 

8 

1  2 

7 

50 

5 

37 

4 

25 

3 

13 

1 

0 

0 

13 

21 

9 

7 

8 

54 

6 

40 

5 

27 

3 

14 

2 

0 

0 

14 

27 

11 

12 

9 

58 

7 

44 

5 

29 

4 

15 

2 

0 

0 

15 

33 

12 

18 

10 

1  2 

8 

47 

6 

31 

4 

16 

2 

0 

0 

16 

39 

14 

23 

12 

6 

9 

50 

7 

33 

5 

17 

2 

0 

0 

17 

45 

16 

28 

13 

10 

10 

53 

8 

35 

5 

18 

3 

0 

0 

18 

51 

18 

33 

15 

14 

12 

56 

9 

37 

6 

19 

3 

0 

0 

19 

57 

20 

38 

16 

18 

13 

59 

10 

39 

7 

20 

4 

0 

0 

20 

2  3 

22 

43 

18 

22 

14 

1  2 

11 

41 

7 

21 

4 

0 

0 

21 

9 

24 

48 

20 

26 

16 

4 

12 

43 

8 

22 

4 

0 

0 

22 

15 

26 

52 

22 

30 

17 

7 

13 

45 

9 

22 

4 

0 

0 

23 

20 

28 

57 

24 

34 

19 

10 

14 

47 

10 

23 

5 

0 

0 

24 

26 

31 

2  2 

26 

38 

21 

13 

16 

49 

10 

24 

5 

0 

0 

25 

32 

34 

7 

28 

41 

22 

16 

17 

51 

11 

25 

6 

0 

0 

26 

38 

36 

11 

30 

45 

24 

19 

18 

53 

12 

26 

6 

0 

0 

27 

43 

39 

16 

33 

49 

26 

22 

20 

54 

13 

27 

7 

0 

0 

28 

49 

42 

21 

35 

53 

28 

24 

21 

56 

14 

28 

7 

0 

0 

29 

54 

45 

25 

37 

56 

30 

27 

23 

58 

15 

29 

8 

0 

0 

30 

3  0 

48 

30 

40 

2  0 

32 

30 

24 

1  0 

16 

30 

8 

0 

0 

31 

5 

51 

35 

43 

4 

34 

33 

26 

2 

17 

31 

9 

0 

0 

32 

11 

54 

39 

45 

7 

36 

35 

27 

4 

18 

32 

9 

0 

0 

33 

16 

58 

43 

48 

11 

39 

38 

29 

5 

19 

33 

10 

0 

0 

34 

21 

85  1 

48 

51 

14 

41 

41 

31 

7 

20 

34 

10 

0 

0 

35 

26 

5 

52 

54 

18 

43 

43 

33 

9 

22 

34 

11 

0 

0 

36 

31 

8 

56 

57 

21 

46 

46 

34 

11 

23 

35 

11 

0 

0 

37 

36 

12 

3  0 

86  0 

24 

48 

48 

36 

12 

24 

36 

12 

0 

0 

38 

41 

16 

5 

3 

28 

51 

51 

38 

14 

25 

37 

13 

0 

0 

39 

46 

20 

9 

7 

31 

53 

53 

40 

16 

27 

38 

14 

0 

0 

40 

51 

24 

13 

10 

34 

56 

56 

42 

17 

28 

39 

14 

0 

0 

41 

56 

28 

17 

13 

37 

59 

58 

44 

19 

29 

39 

15 

0 

0 

42 

4  1 

32 

21 

17 

41 

87  2 

2  0 

46 

20 

31 

40 

15 

0 

0 

43 

5 

36 

25 

20 

44 

4 

3 

48 

22 

32 

41 

16 

0 

0 

44 

10 

41 

28 

24 

47 

7 

5 

50 

23 

34 

42 

17 

0 

0 

45 

14 

45 

32 

28 

50 

10 

7 

53 

25 

35 

43 

18 

0 

0 

Table  13.    Kelvin's  Sumner  Line  Table 


317 


b 

a  =  84° 

a  =  85° 

a  =  86° 

a  =  87° 

a  =  88° 

a  =  89° 

a  =  90° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

45 

414 

8545 

332 

8628 

250 

87  10 

2  7 

8753 

125 

8835 

043 

8918 

0  0 

90  0 

46 

19 

49 

36 

31 

53 

13 

9 

55 

26 

37 

43 

18 

0 

0 

47 

23 

54 

39 

35 

55 

16 

12 

57 

28 

38 

44 

19 

0 

0 

48 

27 

59 

43 

39 

58 

19 

14 

59 

29 

40 

45 

20 

0 

0 

49 

31 

86  3 

46 

43 

3  1 

22 

16 

88  2 

31 

41 

45 

21 

0 

0 

50 

35 

8 

50 

47 

4 

26 

18 

4 

32 

43 

46 

21 

0 

0 

51 

39 

13 

53 

51 

7 

29 

20 

7 

33 

44 

46 

22 

0 

0 

52 

43 

18 

56 

55 

9 

32 

22 

9 

35 

46 

47 

23 

0 

0 

53 

47 

23 

59 

59 

12 

35 

24 

12 

36 

48 

48 

24 

0 

0 

54 

51 

28 

4  2 

87  3 

14 

39 

26 

14 

37 

49 

49 

25 

0 

0 

55 

55 

33 

5 

8 

17 

42 

27 

17 

38 

51 

49 

26 

0 

0 

56 

58 

38 

8 

12 

19 

46 

29 

19 

39 

53 

50 

26 

0 

0 

57 

5  2 

43 

11 

16 

21 

49 

31 

22 

41 

55 

50 

27 

0 

0 

58 

5 

49 

14 

21 

24 

53 

33 

25 

42 

56 

51 

28 

0 

0 

59 

8 

54 

17 

25 

26 

56 

34 

27 

43 

58 

51 

29 

0 

0 

60 

11 

87  0 

20 

30 

28 

88  0 

36 

30 

44 

89  0 

52 

30 

0 

0 

61 

14 

5 

22 

34 

30 

3 

37 

33 

45 

2 

52 

31 

0 

0 

62 

17 

11 

25 

39 

32 

7 

39 

35 

46 

4 

53 

32 

0 

0 

63 

20 

16 

27 

44 

34 

11 

40 

38 

47 

5 

53 

33 

0 

0 

64 

23 

22 

30 

48 

36 

15 

42 

41 

48 

7 

54 

34 

0 

0 

65 

26 

27 

32 

53 

38 

18 

43 

44 

49 

9 

54 

35 

0 

0 

66 

29 

33 

34 

58 

39 

22 

44 

47 

50 

11 

55 

36 

0 

0 

67 

31 

39 

36 

88  3 

41 

26 

46 

50 

50 

13 

55 

37 

0 

0 

68 

34 

45 

38 

7 

42 

30 

47 

53 

51 

15 

56 

38 

0 

0 

69 

36 

51 

40 

12 

44 

34 

48 

55 

52 

17 

56 

38 

0 

0 

70 

38 

56 

42 

17 

46 

38 

49 

58 

53 

:  19 

56 

39 

0 

0 

71 

40 

88  2 

44 

22 

47 

42 

50 

89  1 

53 

21 

56 

40 

0 

0 

72 

42 

8 

45 

27 

48 

46 

51 

4 

54 

23 

57 

41 

0 

0 

73 

44 

14 

47 

32 

50 

50 

52 

7 

55 

25 

57 

42 

0 

0 

74 

46 

20 

49 

37 

51 

54 

53 

10 

55 

27 

58 

43 

0 

0 

75 

48 

27 

50 

42 

52 

58 

54 

13 

56 

29 

58 

44 

0 

0 

76 

49 

33 

51 

47 

53 

89  2 

55 

16 

56 

31 

58 

45 

0 

0 

77 

51 

39 

52 

52 

54 

6 

55 

19 

57 

33 

58 

46 

0 

0 

78 

52 

45 

53 

57 

55 

10 

56 

23 

57 

35 

59 

47 

0 

0 

79 

53 

51 

54 

89  3 

56 

14 

57 

26 

58 

37 

59 

49 

0 

0 

80 

54 

57 

55 

8 

56 

18 

57 

29 

58 

39 

59 

50 

0 

0 

81 

55 

89  3 

56 

13 

57 

22 

58 

32 

59 

41 

59 

51 

0 

0 

82 

56 

10 

57 

18 

58 

27 

58 

35 

59 

43 

59 

52 

0 

0 

83 

57 

16 

58 

23 

58 

31 

59 

38 

59 

45 

1  0 

53 

0 

0 

84 

58 

22 

58 

29 

59 

35 

59 

41 

59 

47 

0 

54 

0 

0 

85 

59 

29 

59 

34 

59 

39 

59 

44 

2  0 

50 

0 

55 

0 

0 

86 

59 

35 

59 

39 

59 

43 

3  0 

47 

0 

52 

0 

56 

0 

0 

87 

6  0 

41 

5  0 

44 

4  0 

47 

0 

51 

0 

54 

0 

57 

0 

0 

88 

0 

47 

0 

50 

0 

52 

0 

54 

0 

56 

0 

58 

0 

0 

89 

0 

54 

0 

55 

0 

56 

0 

57 

0 

58 

0 

59 

0 

0 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

0 

318 


Table  14.    Sumner  Intersection 


LAKGEK 
BEARING 

SMALLER  BEARING 

2° 

4° 

6° 

8° 

10° 

34° 

0.07 

0.14 

0.22 

0.32 

0.43 

36 

.06 

.13 

.21 

.30 

.40 

38 

.06 

.12 

.20 

.28 

.37 

40 

.06 

.12 

.19 

.26 

.35 

42 

.05 

.11 

.18 

.25 

.33 

44 

.05 

.11 

.17 

.24 

.31 

46 

.05 

.10 

.16 

.23 

.30 

48 

.05 

.10 

.16 

.22 

.28 

50 

.05 

.10 

.15 

.21 

.27 

52 

.05 

.09 

.15 

.20 

.26 

54 

.04 

.09 

.14 

.19 

.25 

56 

.04 

.09 

.14 

.19 

.24 

58 

.04 

.09 

.13 

.18 

.23 

60 

.04 

.08 

.13 

.18 

.23 

62 

.04 

.08 

.13 

.17 

.22 

64 

.04 

.08 

.12 

.17 

.21 

66 

.04 

.08 

.12 

.16 

.21 

68 

.04 

.08 

.12 

.16 

.20 

70 

.04 

.08 

.12 

.16 

.20 

72 

.04 

.08 

.11 

.15 

.20 

74 

.04 

.07 

.11 

.15 

.19 

76 

.04 

.07 

.11 

.15 

.19 

78 

.04 

.07 

.11 

.15 

.19 

80 

.04 

.07 

.11 

.15 

.18 

82 

.04 

.07 

.11 

.14 

.18 

84 

.04 

.07 

.11 

.14 

.18 

86 

.04 

.07 

.11 

.14 

.18 

88 

.04 

.07 

.11 

.14 

.18 

90 

.03 

.07 

.11 

.14 

.18 

92 

.03 

.07 

.10 

.14 

.18 

94 

.03 

.07 

.10 

.14 

.17 

96 

.03 

.07 

.10 

.14 

.17 

98 

.04 

.07 

.10 

.14 

.17 

100 

.04 

.07 

.11 

.14 

.17 

102 

.04 

.07 

.11 

.14 

.17 

104 

.04 

.07 

.11 

.14 

.17 

106 

.04 

.07 

.11 

.14 

.17 

108 

.04 

.07 

.11 

.14 

.18 

110 

.04 

.07 

.11 

.14 

.18 

112 

.04 

.07 

.11 

.14 

.18 

114 

.04 

.07 

.11 

.15 

.18 

116 

.04 

.07 

.11 

.15 

.18 

118 

.04 

.08 

.11 

.15 

.18 

120 

.04 

.08 

.11 

.15 

.18 

122 

.04 

.08 

.12 

.15 

.19 

124 

.04 

.08 

.12 

.16 

.19 

126 

.04 

.08 

.12 

.16 

.19 

128 

.04 

.08 

.12 

.16 

.20 

130 

.04 

.09 

.13 

.17 

.20 

132 

.05 

.09 

.13 

.17 

.20 

134 

.05 

.09 

.13 

.17 

.21 

136 

.05 

.09 

.14 

.18 

.21 

138 

.05 

.10 

.14 

.18 

.22 

140 

.05 

.10 

.15 

.19 

.23 

142 

.05 

.10 

.15 

.19 

.23 

144 

.06 

.11 

.16 

.20 

.24 

146 

.06 

.12 

.16 

.21 

.25 

148 

.06 

.12 

.17 

.22 

.26 

150 

.07 

.12 

.18 

.23 

.27 

152 

.07 

.13 

.19 

.24 

•28 

154 

.07 

.14 

.20 

.25 

.30 

156 

.08 

.15 

.21 

.26 

.31 

158 

.09 

.16 

.22 

.28 

.33 

160 

.09 

.17 

.24 

.30 

.35 

LARGER 
BEARING 

SMALLER  BEARING 

12° 

14° 

16° 

18° 

20° 

42° 

0.42 

0.52 

0.63 

0.76 

0.91 

44 

.39 

.48 

.59 

.70 

.84 

46 

.37 

.46 

.55 

.66 

.78 

48 

.35 

.43 

.52 

.62 

.73 

50 

.34 

.41 

.49 

.58 

.68 

52 

.32 

.39 

.47 

.55 

.65 

54 

.31 

.38 

.45 

.52 

.61 

56 

.30 

.36 

.43 

.50 

.58 

58 

.29 

.35 

.41 

.48 

.56 

60 

.28 

.34 

.40 

.46 

.53 

62 

.27 

.33 

.38 

.44 

.51 

64 

.26 

.32 

.37 

.43 

.49 

66 

.26 

.31 

.36 

.42 

.48 

68 

.25 

.30 

.35 

.40 

.46 

70 

.24 

.29 

.34 

.39 

.45 

72 

.24 

.28 

.33 

.38 

.43 

74 

.23 

.28 

.32 

.37 

.42 

76 

.23 

.27 

.32 

.36 

.41 

78 

.23 

.27 

.31 

.36 

.40 

80 

.22 

.26 

.31 

.35 

.39 

82 

.22 

.26 

.30 

.34 

.39 

84 

.22 

.26 

.30 

.34 

.38 

86 

.22 

.25 

.29 

.33 

.37 

88 

.21 

.25 

.29 

.33 

.37 

90 

.21 

.25 

.29 

.32 

.36 

92 

.21 

.25 

.28 

.32 

.36 

94 

.21 

.25 

.28 

.32 

.36 

96 

.21 

.24 

.28 

.32 

.35 

98 

.21 

.24 

.28 

.31 

.35 

100 

.21 

.24 

.28 

.31 

.35 

102 

.21 

.24 

.28 

.31 

.35 

104 

.21 

.24 

.28 

.31 

.34 

106 

.21 

.24 

.28 

.31 

.34 

108 

.21 

.24 

.28 

.31 

.34 

110 

.21 

.24 

.28 

.31 

.34 

112 

.21 

.24 

.28 

.31 

.34 

114 

.21 

.24 

.28 

.31 

.34 

116 

.21 

.25 

.28 

.31 

.34 

118 

.22 

.25 

.28 

.31 

.35 

120 

.22 

.25 

.28 

.32 

.35 

122 

.22 

.25 

.29 

.32 

.35 

124 

.22 

.26 

.29 

.32 

.35 

126 

.23 

.26 

.29 

.32 

.36 

128 

.23 

.26 

.30 

.33 

.36 

130 

.23 

.27 

.30 

.33 

.36 

132 

.24 

.27 

.31 

.34 

.37 

134 

.24 

.28 

.31 

.34 

.37 

136 

.25 

.28 

.32 

.35 

.38 

138 

.26 

.29 

.32 

.36 

.39 

140 

.26 

.30 

.33 

.36 

.39 

142 

.27 

.31 

.34 

.37 

.40 

144 

.28 

.32 

.35 

.38 

.41 

146 

.29 

.33 

.36 

.39 

.42 

148 

.30 

.34 

.37 

.40 

.43 

150 

.31 

.35 

.38 

.42 

.45 

152 

.32 

.36 

.40 

.43 

.46 

154 

.34 

.38 

.41 

.44 

.48 

156 

.35 

.39 

.43 

.46 

.49 

158 

.37 

.41 

.45 

.48 

.51 

160 

.39 

.43 

.47 

.50 

.53 

162 

.41 

.46 

.49 

.52 

.56 

164 

.44 

.48 

.52 

.55 

.58 

166 

.47 

.52 

.55 

.58 

.61 

168 

.50 

.55 

.59 

.62 

.65 

Table  14.    Suniner  Intersection 


319 


LARGER 
BEARING 

SMALLER  BEARING 

22° 

24' 

26° 

28° 

30° 

54° 

0.71 

0.81 

0.93 

1.07 

1.23 

56 

.67 

.77 

.88 

1.00 

1.14 

58 

.64 

.73 

.83 

0.94 

1.07 

60 

.61 

.69 

.78 

.89 

1.00 

62 

.58 

.66 

.75 

.84 

0.94 

64 

.56 

.63 

.71 

.80 

.89 

66 

.54 

.61 

.68 

.76 

.85 

68 

.52 

.59 

.66 

.73 

.81 

70 

.50 

.57 

.63 

.70 

.78 

72 

.49 

.55 

.61 

.68 

.75 

74 

.48 

.53 

.59 

.65 

.72 

76 

.46 

.52 

.57 

.63 

.70 

78 

.45 

.50 

.56 

.61 

.67 

80 

.44 

.49 

.54 

.60 

.65 

82 

.43 

.48 

.53 

.58 

.63 

84 

.42 

.47 

.52 

.57 

.62 

86 

.42 

.46 

.51 

.55 

.60 

88 

.41 

.45 

.50 

.54 

.59 

90 

.40 

.45 

.49' 

.53 

.58 

92 

.40 

.44 

.48 

.52 

.57 

94 

.39 

.43 

.47 

.51 

.56 

96 

.39 

.43 

.47 

.51 

.55 

98 

.39 

.42 

.46 

.50 

.54 

100 

.38 

.42 

.46 

.49 

.53 

102 

.38 

.42 

.45 

.49 

.53 

104 

.38 

.41 

.45 

.48 

.52 

106 

.38 

.41 

.45 

.48 

.52 

108 

.38 

.41 

.44 

.48 

.51 

110 

.37 

.41 

.44 

.47 

.51 

112 

.37 

.41 

.44 

.47 

.50 

114 

.37 

.41 

.44 

.47 

.50 

116 

.38 

.41 

.44 

.47 

.50 

118 

.38 

.41 

.44 

.47 

.50 

120 

.38 

.41 

.44 

.47 

.50 

122 

.38 

.41 

.44 

.47 

.50 

124 

.38 

.41 

.44 

.47 

.50 

126 

.39 

.42 

.45 

.47 

.50 

128 

.39 

.42 

.45 

.48 

.50 

130 

.39 

.42 

.45 

.48 

.51 

132 

.40 

.43 

.46 

.48 

.51 

134 

.40 

.43 

.46 

.49 

.52 

136 

.41 

.44 

.47 

.49 

.52 

138 

.42 

.45 

.47 

.50 

.53 

140 

.42 

.45 

.48 

.51 

.53 

142 

.43 

.46 

.49 

.51 

.54 

144 

.44 

.47 

.50 

.52 

.55 

146 

.45 

.48 

.51 

.53 

.56 

148 

.46 

.49 

.52 

.54 

.57 

150 

.48 

.50 

.53 

.55 

.58 

152 

.49 

.52 

.54 

.57 

.59 

154 

.50 

.53 

.56 

.58 

.60 

156 

.52 

.55 

.57 

.60 

.62 

158 

.54 

.57 

.59 

.61 

.63 

160 

.56 

.59 

.61 

.63 

.65 

162 

.58 

.61 

.63 

.65 

.67 

164 

.61 

.63 

.66 

.68 

.70 

166 

.64 

.66 

.68 

.70 

.72 

168 

.67 

.69 

.71 

.73 

.75 

170 

.71 

.73 

.75 

.76 

.78 

172 

.75 

.77 

.78 

.80 

.81 

174 

.80 

.81 

.83 

.84 

.85 

176 

.85 

.87 

.88 

.89 

.89 

178 

.92 

.93 

.93 

.94 

.94 

LARGER 
BEARING 

SMALLER  BEARING 

32° 

34° 

36° 

38° 

40° 

62° 

1.06 

1.19 

1.34 

1.51 

1.72 

64 

1.00 

1.12 

1.25 

1.40 

.58 

66 

0.95 

1.06 

1.18 

1.31 

.47 

68 

.90 

1.00 

1.11 

1.23 

.37 

70 

.86 

0.95 

1.05 

1.16 

.29 

72 

.82 

.91 

1.00 

1.10 

.21 

74 

.79 

.87 

0.95 

1.05 

.15 

76 

.76 

.84 

.91 

1.00 

1.09 

78 

.74 

.80 

.88 

0.96 

1.04 

80 

.71 

.78 

.85 

.92 

1.00 

82 

.69 

.75 

.82 

.89 

0.96 

84 

.67 

.73 

.79 

.86 

.93 

86 

.66 

.71 

.77 

.83 

.89 

88 

.64 

.69 

.75 

.80 

.86 

90 

.62 

.67 

.73 

.78 

.84 

92 

.61 

.66 

.71 

.76 

.82 

94 

.60 

.65 

.69 

.74 

.79 

96 

.59 

.63 

.68 

.73 

.78 

98 

.58 

.62 

.67 

.71 

.76 

100 

.57 

.61 

.65 

.70 

.74 

102 

.56 

.60 

.64 

.68 

.73 

104 

.56 

.60 

.63 

.67 

.72 

106 

.55 

.59 

.63 

.66 

.70 

108 

.55 

.58 

.62 

.66 

.69 

110 

.54 

.58 

.61 

.65 

.68 

112 

.54 

.57 

.61 

.64 

.68 

114 

.54 

.57 

.60 

.63 

.67 

116 

.53 

.56 

.60 

.63 

.66 

118 

.53 

.56 

.59 

.63 

.66 

120 

.53 

.56 

.59 

.62 

.65 

122 

.53 

.56 

.59 

.62 

.65 

124 

.53 

.56 

.59 

.62 

.65 

126 

.53 

.56 

.59 

.62 

.64 

128 

.53 

.56 

.59 

.62 

.64 

130 

.54 

.56 

.59 

.62 

.64 

132 

.54 

.56 

.59 

.62 

.64 

134 

.54 

.57 

.59 

.62 

.64 

136 

.55 

.57 

.60 

.62 

.65 

138 

.55 

.58 

.60 

.63 

.65 

140 

.56 

.58 

.61 

.63 

.65 

142 

.56 

.59 

.61 

.63 

.66 

144 

.57 

.60 

.62 

.64 

.66 

146 

.58 

.60 

.63 

.65 

.67 

148 

.59 

.61 

.63 

.66 

.68 

150 

.60 

.62 

.64 

.66 

.68 

152 

.61 

.63 

.65 

.67 

.69 

154 

.62 

.65 

.67 

.68 

.70 

156 

.64 

.66 

.68 

.70 

.72 

158 

.66 

.67 

.69 

.71 

.73 

160 

.67 

.69 

.71 

.73 

.74 

162 

.69 

.71 

.73 

.74 

.76 

164 

.71 

.73 

.75 

.76 

.78 

166 

.74 

.75 

.77 

.78 

.79 

168 

.76 

.78 

.79 

.80 

.82 

170 

.79 

.80 

.82 

.83 

.84 

172 

.82 

.84 

.85 

.86 

.86 

174 

.86 

.87 

.88 

.89 

.89 

176 

.90 

.91 

.91 

.92 

.93 

178 

.95 

.95 

.95 

.96 

.96 

320 


Table  14.    Simmer  Intersection 


LARGER 
BEARING 

SMALLER  BEARING 

42° 

44° 

46° 

48° 

50° 

72° 

1.34 

1.48 

1.64 

1.83 

2.04 

74 

1.26 

1.39 

1.53 

1.70 

1.88 

76 

1.20 

1.31 

1.44 

1.58 

1.75 

78 

1.14 

1.24 

1.36 

1.49 

1.63 

80 

1.09 

1.18 

1.28 

1.40 

1.53 

82 

1.04 

1.13 

1.22 

1.33 

1.45 

84 

1.00 

1.08 

1.17 

1.26 

1.37 

86 

0.96 

1.04 

1.12 

1.21 

1.30 

88 

.93 

1.00 

1.08 

1.16 

1.24 

90 

.90 

0.97 

1.04 

1.11 

.19 

92 

.87 

.93 

1.00 

1.07 

.14 

94 

.85 

.91 

0.97 

1.03 

.10 

96 

.83 

.88 

.94 

1.00 

.06 

98 

.81 

.86 

.91 

0.97 

.03 

100 

.79 

.84 

.89 

.94 

.00 

102 

.77 

.82 

.87 

.92 

0.97 

104 

.76 

.80 

.85 

.90 

.95 

106 

.74 

.79 

.83 

.88 

.92 

108 

.73 

.77 

.81 

.86 

.90 

110 

.72 

.76 

.80 

.84 

.88 

112 

.71 

.75 

.79 

.83 

.87 

114 

.70 

.74 

.78 

.81 

.85 

116 

.70 

.73 

.77 

.80 

.84 

118 

.69 

.72 

.76 

.79 

.83 

120 

.68 

.72 

.75 

.78 

.82 

122 

.68 

.71 

.74 

.77 

.81 

124 

.68 

.71 

.74 

.77 

.80 

126 

.67 

.70 

.73 

.76 

.79 

128 

.67 

.70 

.73 

.75 

.78 

130 

.67 

.70 

.72 

.75 

.78 

132 

.67 

.70 

.72 

.75 

.77 

134 

.67 

.69 

.72 

.74 

.77 

136 

.67 

.70 

.72 

.74 

.77 

138 

.67 

.70 

.72 

.74 

.77 

140 

.68 

.70 

.72 

.74 

.77 

142 

.68 

.70 

.72 

.74 

.77 

144 

.68 

.71 

.73 

.75 

.77 

146 

.69 

.71 

.73 

.75 

.77 

148 

.70 

.72 

.74 

.76 

.77 

150 

.70 

.72 

.74 

.76 

.78 

152 

.71 

.73 

.75 

.77 

.78 

154 

.72 

.74 

.76 

.77 

.79 

156 

.73 

.75 

.77 

.78 

.80 

158 

.74 

.76 

.78 

.79 

.81 

160 

.76 

.77 

.79 

.80 

.82 

162 

.77 

.79 

.80 

.81 

.83 

164 

.79 

.80 

.81 

.83 

.84 

166 

.81 

.82 

.83 

.84 

.85 

168 

.83 

.84 

.85 

.86 

.87 

170 

.85 

.86 

.87 

.88 

.88 

172 

.87 

.88 

.89 

.90 

.90 

174 

.90 

.91 

.91 

.92 

.92 

176 

.93 

.93 

.94 

.94 

.95 

178 

.96 

.97 

.97 

.97 

.97 

LARGER 
BEARING. 

SMALLER  BEARING 

52° 

54° 

56° 

58° 

60° 

82° 

1.58 

1.72 

1.89 

2.08 

2.31 

84 

1.49 

1.62 

1.77 

.93 

2.13 

86 

1.41 

1.53 

1.66 

.81 

1.98 

88 

1.34 

1.45 

1.56 

.70 

1.84 

90 

1.28 

1.38 

1.48 

.60 

1.73 

92 

1.23 

1.31 

1.41 

.52 

1.63 

94 

1.18 

1.26 

1.35 

1.44 

1.55 

96 

1.13 

1.21 

1.29 

1.38 

1.47 

98 

1.10 

1.16 

1.24 

1.32 

1.41 

100 

1.06 

1.12 

.19 

1.27 

1.35 

102 

1.03 

1.09 

.15 

1.22 

1.29 

104 

1.00 

1.06 

.12 

1.18 

1.25 

106 

0.97 

1.03 

.09 

1.14 

1.20 

108 

.95 

1.00 

.05 

1.11 

1.17 

110 

.93 

0.98 

1.02 

1.08 

1.13 

112 

.91 

.95 

1.00 

1.05 

1.10 

114 

.89 

.93 

0.98 

1.02 

1.07 

116 

.88 

.92 

.96 

1.00 

1.04 

118 

.86 

.90 

.94 

0.98 

1.02 

120 

.85 

.89 

.91 

.96 

1.00 

122 

.84 

.87 

.90 

.95 

0.98 

124 

.83 

.86 

.90 

.93 

.96 

126 

.82 

.85 

.88 

.91 

.95 

128 

.81 

.84 

.87 

.90 

.93 

130 

.81 

.83 

.86 

.89 

.92 

132 

.80 

.83 

.85 

.88 

.91 

134 

.80 

.82 

.85 

.87 

.90 

136 

.80 

.82 

.84 

.87 

.89 

138 

.79 

.81 

.84 

.86 

.89 

140 

.79 

.81 

.83 

.86 

.88 

142 

.79 

.81 

.83 

.85 

.87 

144 

.79 

.81 

.83 

.85 

.87 

146 

.79 

.81 

.83 

.85 

.87 

148 

.79 

.81 

.83 

.85 

.87 

150 

.80 

.81 

.83 

.85 

.87 

152 

.80 

.82 

.83 

.85 

.87 

154 

.81 

.82 

.84 

.85 

.87 

156 

.81 

.83 

.84 

.86 

.87 

158 

.82 

.83 

.85 

.86 

.87 

160 

.83 

.84 

.85 

.86 

.88 

162 

.84 

.85 

.86 

.87 

.89 

164 

.85 

.86 

.87 

.88 

.89 

166 

.86 

.87 

.88 

.89 

.90 

168 

.88 

.89 

.90 

.90 

.91 

170 

.89 

.90 

.90 

.91 

.92 

172 

.91 

.92 

.91 

.93 

.93 

174 

.93 

.93 

.94 

.95 

.95 

176 

.95 

.95 

.96 

.96 

.96 

178 

.97 

.98 

.98 

.98 

.98 

Table  14.    Sumner  Intersection 


321 


LARGER 
BEARING 

SMALLER  BEARING 

62° 

64° 

66° 

68° 

70° 

92° 

1.77 

1.91 

2.08 

2.28 

2.51 

94 

.67 

1.80 

1.95 

2.12 

2.31 

96 

.58 

1.70 

.83 

1.97 

2.14 

98 

.50 

1.61 

.72 

1.85 

2.00 

100 

.43 

1.53 

.63 

.75 

1.88 

102 

.37 

1.46 

.55 

.66 

1.77 

104 

.32 

1.40 

.48 

.58 

1.68 

106 

.27 

1.34 

.42 

.51 

1.60 

108 

.23 

1.29 

.37 

.44 

1.53 

110 

.19 

1.25 

.32 

.39 

1.46 

112 

.15 

1.21 

.27 

.33 

1.40 

114 

.12 

1.17 

.23 

.29 

1.35 

116 

.09 

1.14 

.19 

.25 

1.31 

118 

.07 

1.11 

.16 

.21 

1.26 

120 

1.04 

1.08 

.13 

.18 

1.23 

122 

1.02 

1.06 

.10 

.15 

1.19 

124 

1.00 

1.04 

.08 

.12 

1.16 

126 

0.98 

1.02 

.05 

1.09 

1.13 

128 

.97 

1.00 

.03 

1.07 

1.11 

130 

.95 

0.98 

.02 

1.05 

1.09 

132 

.94 

.97 

.00 

1.03 

1.06 

134 

.93 

.96 

0.99 

1.01 

1.04 

136 

.92 

.95 

.97 

1.00 

1.03 

138 

.91 

.94 

.96 

0.99 

1.01 

140 

.90 

.93 

.95 

.97 

1.00 

142 

.90 

.92 

.94 

.96 

0.99 

144 

.89 

.91 

.93 

.96 

.98 

146 

.89 

.91 

.93 

.95 

.97 

148 

.89 

.90 

.92 

.94 

.96 

150 

.88 

.90 

.92 

.94 

.95 

152 

.88 

.90 

.92 

.93 

.95 

154 

.88 

.90 

.91 

.93 

.94 

156 

.89 

.90 

.91 

.93 

.94 

158 

.89 

.90 

.91 

.93 

.94 

160 

.89 

.90 

.91 

.93 

.94 

162 

.90 

.91 

.92 

.93 

.94 

164 

.90 

.91 

.92 

.93 

.94 

166 

.91 

.92 

.93 

.93 

.94 

168 

.92 

.93 

.93 

.94 

.94 

170 

.93 

.94 

.94 

.95 

.95 

172 

.94 

.95 

.95 

.96 

.96 

174 

.95 

.96 

.96 

.96 

.97 

176 

.97 

.97 

.97 

.97 

.98 

178 

.98 

.98 

.99 

.99 

.99 

LARGER 
BEARING 

SMALLER  BEARING 

72° 

74° 

76° 

78° 

80° 

102° 

1.90 

2.05 

2.21 

2.40 

2.63 

104 

1.79 

1.92 

2.07 

2.23 

2.42 

106 

1.70 

1.81 

1.94 

2.08 

2.25 

108 

1.62 

1.72 

1.83 

1.96 

2.10 

110 

1.54 

1.64 

1.74 

1.85 

.97 

112 

1.48 

1.56 

1.65 

1.75 

.86 

114 

1.42 

1.50 

1.58 

1.66 

.76 

116 

1.37 

1.44 

1.51 

1.59 

.68 

118 

1.32 

1.38 

1.45 

1.52 

.60 

120 

1.28 

1.34 

1.40 

1.46 

.53 

122 

1.24 

1.29 

1.35 

1.41 

.47 

124 

1.21 

1.25 

1.31 

1.36 

.42 

126 

1.18 

1.22 

1.27 

1.32 

.37 

128 

1.15 

1.19 

1.23 

1.28 

.33 

130 

1.12 

1.16 

1.20 

1.24 

.29 

132 

1.10 

1.13 

1.17 

1.21 

.25 

134 

1.08 

1.11 

1.14 

1.18 

.22 

136 

1.06 

1.09 

1.12 

1.15 

.19 

138 

1.04 

1.07 

1.10 

1.13 

.16 

140 

1.03 

1.05 

1.08 

1.11 

.14 

142 

1.01 

1.04 

1.06 

.09 

.12 

144 

1.00 

1.02 

1.05 

.07 

.10 

146 

0.99 

1.01 

1.03 

.05 

.08 

148 

.98 

1.00 

1.02 

.04 

.06 

150 

.97 

0.99 

1.01 

.03 

.05 

152 

.97 

.98 

1.00 

.02 

.04 

154 

.96 

.98 

0.99 

.01 

.02 

156 

.96 

.97 

.99 

.00 

.01 

158 

.95 

.97 

.98 

0.99 

.01 

160 

.95 

.96 

.98 

.99 

.00 

162 

.95 

.96 

.98 

.99 

.00 

164 

.95 

.96 

.98 

.99 

0.99 

166 

.96 

.96 

.98 

.99 

.99 

168 

.96 

.96 

.98 

.99 

.99 

170 

.97 

.97 

.98 

.99 

.99 

172 

.97 

.97 

.99 

.99 

.99 

174 

.98 

.98 

.99 

.99 

.99 

176 

.99 

.98 

.99 

.99 

.99 

178 

.99 

.99 

.99 

.99 

.99 

322 


Table  14.    Simmer  Intersection 


LARGER 
BEARIN 

•  SMALLER  BEARING 

82° 

84° 

86° 

88° 

90° 

112° 

1.98 

2.12 

2.28 

2.46 

2.67 

114 

1.87 

1.99 

2.12 

2.28 

2.46 

116 

1.77 

1.88 

2.00 

2.13 

2.28 

118 

1.68 

1.78 

1.88 

2.00 

2.13 

120 

1.61 

1.69 

1.78 

1.89 

2.00 

122 

1.54 

1.62 

1.70 

.79 

1.89 

124 

1.48 

1.55 

1.62 

.70 

1.79 

126 

1.43 

1.48 

1.55 

.62 

1.70 

128 

1.38 

1.43 

1.49 

.55 

1.62 

130 

1.33 

1.38 

1.44 

.49 

1.56 

132 

1.29 

1.34 

1.39 

1.44 

1.49 

134 

1.26 

1.30 

1.34 

1.39 

1.44 

136 

1.22 

1.26 

1.30 

1.34 

1.39 

138 

1.19 

1.23 

1.27 

1.30 

1.35 

140 

1.17 

1.20 

1.23 

1.27 

1.31 

142 

1.14 

1.17 

1.20 

1.24 

1.27 

144 

1.12 

1.15 

1.18 

1.21 

1.24 

146 

1.10 

1.13 

1.15 

1.18 

1.21 

148 

1.08 

1.11 

1.13 

1.15 

1.18 

150 

1.07 

1.09 

1.11 

1.13 

1.15 

152 

1.05 

1.07 

1.09 

1.11 

1.13 

154 

1.04 

1.06 

1.08 

1.09 

1.11 

156 

1.03 

1.05 

1.06 

1.08 

1.09 

158 

1.02 

1.03 

1.05 

1.06 

1.08 

160 

1.01 

1.02 

1.04 

1.05 

1.06 

162 

1.00 

1.01 

1.03 

1.03 

.05 

164 

.00 

1.00 

1.02 

1.02 

.04 

166 

.00 

.00 

1.01 

1.02 

.03 

168 

.00 

.00 

.00 

1.01 

.02 

170 

0.99 

.00 

.00 

.00 

.01 

172 

.99 

.00 

.00 

.00 

.01 

174 

.99 

0.99 

.00 

.00 

.00 

176 

.99 

.99 

.99 

.00 

.00 

178 

.99 

.99 

.99 

0.99 

.00 

INDEX 


Accuracy,  how  attained  in  Sumner 
method,  137 

in  interpolating  declination,  82 

in  reading  sextant,  66,  73 
Address,  of  house  in  city,  2 

of  ship  at  sea,  3 
Adjuster,  compass,  43 

deviation  table,  48 
Adjusting  screws,  68 
Adjustments  of  sextant,  68 
Advance    preparation    for    observa- 
tion, star,  96 

sun,  92 
"Afternoon"     observation    of    star, 

120 

Aid  to  navigation,  5. 
Almanac,  nautical,  chapter  on,  75 

specimen  pages,  76,  83,  84,  91,  92, 

97,  98 

Altitude,  changes  slowly  near   noon, 
88 

denned,  61 

maximum  and  minimum,  89 

maximum  of  star,  90 

sun  or  star,  correction  of,  70 

used  for  noon-sight,  86 
A.M.  and  P.M.,  75 
American  ephemeris,  75 
Angle,  auxiliary,  in  Sumner  method, 
112 

auxiliary,  in  azimuth  table,  115 

course,  10 

danger,  58 

deck,  59 

denned,  8 

how  designated,  8 

how  measured,  9 

of  triangle,  31 

pairs  of,  for  bow  bearings,  56 

vertex  of,  8 
Apparent  solar  day,  77 

time,  77 

time,  used  for  ship's  clock,  94 
Arc,  minutes  of,  3 


Argument,  difference,  12,  27 

double,  in  haversine  table,  100 

in  tables,  10 

pairs  of,  11 

pairs  of,  in  log  table,  25 

pairs  of,  in  traverse  table,  14 

pairs    of,    in    trigonometric    table, 

32 

Arrow  on  sextant  vernier,  65 
Artificial  horizon,  69 

accuracy  of,  73 

Ashore,  observing  with  sextant,  69 
Astronomic  time,  75,  78 
Auxiliary,   angle,   in   azimuth   table, 
115 

angle,  in  Sumner  method,  112 

azimuth  table,  115 
Avenues    and    streets     imagined    by 

navigator,  3 
Azimuth,  circle  (instrument),  44,  53 

how  counted,  114 

of  sun,  defined,  44 

table,  111,  113 

table,  auxiliary,  115 

Beam,  bearing,  55 
Bearing,  bow  and  beam,  55 

compass,  of  sun,  44 

cross,  56 

danger,  59 

favorable,  for  time-sight,  99 

from  the  bow,  doubling  it,  54 

from  the  bow,  in  Sumner  naviga- 
tion, 134 

of  objects  ashore,  53 

of  Sumner  line,  111 

reversed,  55 

Below  pole  observations,  89 
Bow,  and  beam  bearings,  55 

bearings  from  the,  doubling,  54 

bearings  from  the,  use  in  Sumner 

navigation,  134 
Bowl  of  compass,  42 
Boxing  the  compass,  41 


323 


324 


INDEX 


Cabin  clock,  how  set,  94 

Calipers,  used  in  testing  sextant,  69 

Card,  compass,  41 

rate,  of  chronometer,  79 
Cardinal  points  of  compass,  41 
Celestial,  equator,  85 

Greenwich,  85 

poles,  85 

sphere,  85 
Changed    D.    R.    point    in    Sumner 

navigation,  126 
Chart,  great  circle  sailing,  37 

how  to  draw  Sumner  lines  on,   123 

in  dead  reckoning,  8 

laying  down  ship's  place  on,  55 

Mercator,  35 

Mercator,  meridians  on,  38 

projections,  38 

scale  of,  55 
Chronometer,  6 

dial,  divided  in  12  hours,  93 

error,  79 

face,  93 

keeps  mean  solar  time,  77 

minus  watch,  94 

rated,  79 

used  in  interpolating  declination, 

90 

Circle,  graduated,  of  sextant,  61 
Civil  day,  77 

time,  78,  93 
Clamp  of  sextant,  62 
Clock,  deck  and  cabin,  how  set,  94 
Clockwise,  numbering  of  compass,  41 
Coastwise  navigation,  chapter  on,  53 
Compass,  adjuster,  43 

bearing  of  objects  ashore,  53 

bearing  of  sun,  44 

bowl,  42 

boxing,  41 

card,  41 

chapter  on,  40 

compensating,  43 

course  and  true  course,  42 

deviation  by  observation,  44,  115 

deviation  table,  48 

error  of,  43 

error  of,  when  negative,  47 

formulas,  45 

gyro,  42 

lubber  line,  42 

north,  43 

old-fashioned  designation  of  error, 
47 


Compass,  points  of,  10,  15,  52 

points,  cardinal  and  inter-cardinal, 
41 

variation,  43 

Compensating  compass,  43 
Composite  sailing,  39 
Correction,  of  altitudes,  70 

tables,  for  altitudes,  72 
Cosecant,  31 
Cosine,  31 

-haversine    formulas,    in    Sumner 

method,  112 
Cotangent,  31 
Course,  angle,  10 

magnetic,  49 

protractor,  55 

ship's,  defined,  8 

ship's,  error  of  compass,  43 

ship's,  in  points,  15 

ship's,  measured,  9,  19 

ship's,  old  way  of  measuring,   19, 
22 

ship's,  shaping  it,  44 

ship's,  true  and  compass,  42 

steering,  51 
Cross-bearings,  56 
Currents,  ocean,  effect  of,  2 

ocean,  measured,  103 

Daily  rate  of  chronometer,  79 
Danger,  angle,  57 

bearing,  59 

Date,  right,  in  almanac,  81 
Day,  apparent  solar,  77 

at  sea,  navigator's,  141 

civil,  77 

mean  solar,  77 

solar,  75 
Dead  reckoning,  begins,  53 

chapter  on,  7,  23 

defined,  5 

fundamental  problems  of,  8,  10,  20 

point,  changed,  in  Sumner  naviga- 
tion, 126 

point,  in  Sumner  navigation,  112 

with  logs,  23,  31,  33 
Decimal  part  of  log,  23 

rule  for,  25 
Deck,  angle,  59 

clock,  how  set,  94 

Declination,     interpolated     without 
chronometer,  90 

north  and  south,  plus  and  minus, 


INDEX 


325 


Declination,  of  star,  85,  89,  92 

of  star,  slow  change  of,  90 

of  sun,  75,  82,  85 

of  sun,  minus,  87 

Degrees,  of  latitude    and  longitude, 
when  equal,  4 

of  longitude,  3 

of   longitude,    counted    East    and 
West,  3 

on  compass  card,  41 

subdivisions  of,  3 

unit  for  measuring  angles,  9 
Departure,  denned,  10 

new,  2 

port  of,  1 

relation  of  longitude  difference  to, 
16,  20 

summed,  39 

taking  a,  53 
Destination,  port  of,  1 
Deviation,  of  compass,  43 

of  compass,  by  observation,  44,  115 

of  compass,  older  designation  of,  47 

of  compass,  when  negative,  47 

table,  48,  49 

table,  inverse  use  of,  48 

table,  second,  49 

Dial,    chronometer,    divided    in    12 
hours,  93 

of  compass,  41 
Difference,  argument,  12,  27 

hourly,  79 

interpolation,  28 

latitude,  denned,  10 

latitude,  in  miles  and  minutes,  15 

latitude,  meridional,  36 

latitude,  summed,  39 

longitude,    converted   into    depar- 
ture, 15,  19 

tabular,  12,  27 

time,  80 

Digits,  denned,  24 
Dip,  of  sea  horizon,  70 

of  sea  horizon,  errors  in,  73 

of  sea  horizon,  table  of,  73 

of  star,  at  upper  transit,  96 

of  sun,  at  noon-sight,  86 
Direction,  denned,  8 

difference  of,  8 
Disk  of  sun  in  telescope,  67 
Distance,  in  dead  reckoning,  10 
in  traverse  table,  13 
laying  off  on  chart,  55 
polar,  100 


Dividend,  log  of,  30 

Division,  by  logs,  30 

Divisor,  log  of,  30 

Documents,  superintendent  of,  75 

Double,  argument  in  haversine  table, 

100 

interpolation,  12 
Doubling  the  bearing  from  the  bow, 

54 

Ephemeris,  American,  75 
Equation  of  time,  75,  77 
Equator,  celestial,  85 

earth's,  a  circle,  4 

longitude  meridians  begin  at,  3 
Equinox,  vernal,  85 
Error,  index,  of  sextant,  66 

in  tabular  dip  of  horizon,  73 

of  chronometer,  79 

of  compass,  43 

of  compass,  older  designation  of,  47 

of  compass,  when  negative,  47 
Exactness  of  logs,  30 
Ex-meridian  observation,  99 

Face,  chronometer,  93 

Factor,  correction,  in  Table  2,  17 

log  of,  30 

Favorable  bearing  for  time-sight,  99 
"Fix"  denned,  53 

"Forenoon"  observation  of  star,  120 
Formulas,   cosine-haversine,  for  alti- 
tude, 112 

for  compass,  45 

for  dead  reckoning,  33,  34 

for  latitude  by  noon-sight,  87,  89 

for  magnetic  course,  49 

for  Mercator  sailing,  36 

for  operations  by  Kelvin's  Sumner 
table,  127 

for  Sumner  intersection  point,  135 

for  time-sight,  100 

for  time-sight,  star,  105 

Gaining  rate  of  chronometer,  79 
Graduated  circle,  of  sextant,  61 

off  arc,  66 
Great  circle,  37 

chart,  38 

sailing,  35,  37 
Greenwich,  beginning  of  longitude,  3 

celestial,  85 

mean  time,  77 
Gyro-compass,  42 


326 


INDEX 


Half  sum,  100 

Haversines,  99 

High  latitudes,  noon-sight  in,  89 

Horizon,  61 

artificial,  69 

dip  of,  70 

glass,  on  sextant,  64 
Hour-angle  of  star,  104,  120 
Hourly  difference,  79 

Imaginary  ship,   in  Sumner  naviga- 
tion, 134 
Index,  error,  of  sextant,  66 

error,  of  sextant,  from  sun  obser- 
vations, 67 

glass,  of  sextant,  64 

number,  in  azimuth  table,  114 

of  sextant,  62 

Initial  meridian,  Greenwich,  3 
Inter-cardinal  points  of  compass,  41 
Interpolation,  12 

difference,  28 

double,  12 

in  trigonometric  tables,  33 

inverse,  29 

of  logs,  27 

Intersection  point  of  Sumner  lines, 
111,  125 

finding  it,  133 
Inverse  interpolation,  29 

use  of  deviation  table,  48 

use  of  mathematical  tables,  12 

use  of  Table  3,  28 

use  of  Table  4,  33 
Inverting  telescope,  72 

Kelvin,  Lord,  his  table,  126 

improves  Summer's  method,  112 

Land-marks,  observed  from  ship,  53 
Lane  route,  95 

Latitude,  degrees  of,  equal  to  degrees 
of  longitude,  4 

determined  by  noon-sight,  85 

determined  by  sextant,  5 

determined  by  star,  89 

difference,  defined,  10 

difference,  in  miles  and  minutes,  15 

difference,  meridional,  36 

difference,  summed,  39 

for  time-sight  calculation,  101 

high,  noon-sight  in,  89 

parallels  of,  defined,  3 


Laying  down  ship's  place  on  chart,  55 

off  distance  on  chart,  55 
Lead-line,  60 
Light-gathering  power    of  telescope, 

63 
Lighthouse,  aid  to  navigation,  5 

observed  from  ship,  53 
Limb,  lower,  of  sun,  71 
Line  of  soundings,  60 

Sumner,  or  line  of  position,  109 

Sumner,  principle  of,  111 
Log,  abbreviation  for  logarithm,  23 

instrument  for  estimating  speed  of 

ship,  54 

Logarithms,    decimal   part   of,    rule 
for,  25 

exactness  of,  30 

in  dead  reckoning,  chapter  on,  23 

interpolation  of,  27 

multiplication   and   division   with, 
30 

of  multiples  of  10,  24 

of  products,  30 

table  of,  25 

table  of,  inverse  use  of,  28 

table  of,  trigonometric,  32 

two  parts  of,  23 

whole  number  part  of,  rule  for,  24 
Longitude,  begins  at  Greenwich,  3 

degrees  of,  3 

degrees  of,  counted  east  and  west, 
3 

degrees  of,  when  equal  to  degrees 
of  latitude,  4 

determined  from  star,  104 

determined  from  sun,  99 

difference,  converted    into    depar- 
ture, 15,  19 

difference,    converted    into    time- 
difference,  81 

meridians  of,  defined,  3 

minutes  and  seconds  of,  4 

not  determinate  by  noon-sight,  88 
Losing  rate  of  chronometer,  79 
Lower,  limb  of  sun,  71,  86 

transit,  89 
Lubber-line  of  compass,  42 

Magnetic  course,  49 
Maps,  lines  on,  4 
Mathematical  tables,  10 

inverse  use  of,  12 
Maximum,  altitude,  89 

of  star,  90 


INDEX 


327 


Mean,  of  observations,  73 

solar  day,  77 

solar  time,  77 

sun,  83 

time,  Greenwich,  77 
Mercator,  charts,  35 

charts,  meridians  on,  38 

Gerhard,  35 

sailing,  35 

Mercury,  in  artificial  horizon,  69 
Meridian,  how  numbered,  3 

initial,  at  Greenwich,  3 

of  longitude,  defined,  3 

on  maps,  4 

on  Mercator  chart,  35 

observation,  86 

transit,  96 
Meridional,  latitude  difference,  36 

parts,  35 
Middle  latitude,  16 

sailing,  35 

sailing,  formulas  for,  33,  34 
Midnight  sun,  observed  for  latitude, 

89 

Miles,  nautical,  15 
Minimum,  altitude,  89 
Minus,  declination,  82,  87 

defined,  25 

sign,  for  south  latitude  and  decli- 
nation, 89 

sign,  in  formula,  87 
Minuses  of  arc,  3 

subdivided,  4 

used  in  measuring  angles,  9 
Mirror,  artificial  horizon,  69 

sextant,  61 

Moon,  observation  of,  99 
Motion,  of  ship,  between  two  Sum- 

ner  observations,  137 
Multiples  of  10,  logs  of,  24 
Multiplication,  by  logarithms,  30 

table,  11,  28 

Nautical  almanac,  chapter  on,  75 

miles,  15 

specimen  pages,  76,  83,  84,  91,  92, 

97,  98 

Navigator,  his  day  at  sea,  141 
Negative,  compass  error,  47 

defined,  24 
Newer  navigation  methods,  chapter 

on,  108 
Noon,  75 

sight,  86 


Noon,  sight,  advance  preparation  for, 
92 

sight,  good  for  latitude  only,  88 

sight,  in  high  latitudes,  89 

sight,  in  tropics,  89 

sight,  of  star,  90 
North,  by  compass,  43 

declination  and  latitude  are  plus, 

891 
Numbers,  of  longitude  meridians,  3 

tabular,  10 

Objections  to  Sumner  line,  1101 
Observation,     advance     preparation 

for  sun,  92 

advance  preparation  for  star,  96 
determines  ship's  position,  4 
for  compass  deviation,  44 
planets  and  moon,  98 
single,  determines  one  thing  only, 

88 
Ocean  currents,  deflect  ship,  2 

measured,  103 

Off  arc,  graduations  of  sextant,  66 
Older   navigation   methods,    chapter 
on,  86 

Pages  from  nautical  almanac,  76,  83, 

84,  91,  92,  97,  98 
Pairs,  of  angles  for  bow  bearings,  56 

of  arguments,  11 

of  arguments,  in  log  table,  25 

of  arguments,  in  traverse  table,  14 
Parallax,  correction  of  sun's  altitude, 

71 
Parallel,  of  latitude,  defined,  3 

of  latitude,  how  counted,  3 

of  latitude,  on  maps,  4 

rulers,  55 

sailing,  39 
Parts,  meridional,  35 

of  triangle,  31 

proportional,  28 

two,  of  compass  error,  43 

two,  of  log,  23 

two,  of  log,  rule  for,  24 
Patent  log,  54 
Pelorus,  44,  53 
Plane,  defined,  9 

sailing,  9 

trigonometry,  10 

Planets,  observation  of,  for  latitude, 
98 

observation  of,  time-sight,  104 


328 


INDEX 


Plotting  ship's  place  on  chart,  55 
Plus,  sign  in  formula,  87 

sign,    north   declination   and   lati- 
tude, 89 

P.M.  and  A.M.,  75 

Point,  D.  R.,  in  Sumner  navigation, 
112 

intersection,  of  Sumner  lines,  111, 
125 

Sumner,  112 

Sumner,  ship  not  at,  125 
Points  of  compass,  10,  15 

cardinal,  41 

inter-cardinal,  41 

table  of,  52 
Polar  distance,  100 
Pole,  celestial,  85 

celestial,  observing  below  it,  89 

end  of  longitude  meridians,  3 
Port  of  departure  and  destination,  1 
Position,  line  of,  109 

of  ship  at  sea,  defined,  2 

of  ship,  determined  daily,  2 

of  ship,  how  determined,  4 

of    ship,    most    probable,    not    at 

Sumner  point,  125 
Preparation  for  observation,  star,  96 

sun,  92 

Principle  of  Sumner  line,  111 
Probable    position    of    ship,    not    at 

Sumner  point,  125 

Problem,  of   dead   reckoning,  8,   10, 
20 

of  dead  reckoning,  with  logarithms, 
23,  31,  33 

of  navigation,  1 
Product,  log  of,  30 
Projection  chart,  38 
Proportional  parts,  28 
Protractor,  instrument,  55,  58 

Quotient,  log  of,  30 

Rate  of  chronometer,  79 
Reading  sextant  circle,  62 
Refraction  of  light,  71 
Reversed  bearing,  55 
Rhumb  line,  38 
Right-ascension,  of  star,  85,  89,  91 

of  star,  slow  change  of,  90 

of  sun,  83 

Roof,  of  artificial  horizon,  69 
Rulers,  parallel,  55 


Sailing,  composite,  39 

Mercator,  35 

middle  latitude,  35 

parallel,  39 

plane,  9 

traverse,  39 
Saint      Hilaire,      Marcq,      improves 

Sumner's  method,  112 
Scale  of  chart,  55 
Screw  adjusting,  68 
Secant,  31 

Second  deviation  table,  49 
Seconds  of  arc,  4 
Semi-diameter  of  sun,  71 
Set  of  current  measured,  103 
Setting  watch  on  board,  94 
Sextant,  adjustments,  68 

chapter  on,  61 

clamp,  62 

defined,  5 

determines  latitude,  5 

for  danger  angle,  59 

graduated  circle  of,  61 

graduations  off  arc,  66 

horizon  glass  of,  64 

index,  62 

index  glass,  64 

mirrors,  62 

read  to  minutes  only,  66,  73 

superposed  objects,  64 

tangency  of  sun's  image,  67,  70 

tangent  screw,  62 

telescope,  62 

use  of,  ashore,  69 

vernier,  64 

Shaping  a  course,  39,  44 
Ship,  course,  defined,  8 

course,  measured,  9,  19 

course,  measured,  old  way,  19 

imaginary,  in  Sumner  navigation, 
134 

motion   of,    between   two   Sumner 
observations,  137 

position  of,  how  determined,  4 

position  of,  laying  down  on  chart, 
55 

swinging,  for  compass  adjustment, 
43 

time  is  apparent  solar  time,  94 
Sidereal  time,  85 
Sides  of  triangle,  31 
Sine,  31 

Single   observation     determines   one 
thing  only,  88,  111,  125 


INDEX 


329 


Sky,  85 

Slow-motion  screw  of  sextant,  62 

Solar  day,  75 

apparent,  77 

mean,  77 
Solar  time,  75 

apparent,  77 

mean,  77 
Soundings,  59 

machine,  60 
South      declination      and      latitude, 

minus,  89 

Specimen      pages,      from      nautical 
almanac,  76,  83,  84,  91,  92,  97,  98 
Sphere,  celestial,  85 
Star, .  advance  preparation  for  obser- 
vation, 96 

dip  of,  96 

"forenoon"  and   "afternoon"  ob- 
servation, 120 

hour-angle  of,  104 

"noon-sight"  of,  90 

observed  for  index  error,  67 

observed  for  latitude,  89 

observed  for  Sumner  line,  120 

observed  for  time-sight,  104 

right-ascension  and  declination,  85, 
91,  92 

time  of  upper  transit,  96 
Steering  courses,  table  of,  51 
Streets    and    avenues,    imagined    by 

navigator,  3 
Subdivisions,  of  degrees,  etc.,  3 

of  minutes,  4 

Summed  latitude  difference  and  de- 
parture in  traverse  sailing,  39 
Sumner,  Capt.  Thos.  H.,  109 
Sumner  intersection  table,  135 
Sumner  line,  109 

bearing  of,  111 

how  to  draw  it  on  chart,  123 

objections  to  it,  110 

point  of  intersection,  111,  125 

point  of  intersection,  finding  it,  133 

principle  of,  111 

star  observed  for,  120 

sun  observed  for,  112 
Sumner  method,  auxiliary  angles  in, 
112 

conditions  for  accuracy  in,  137 

cosine-haversine  formula,   112 

with  special  tables,  126 
Sumner   navigation,   compared   with 
time-sight,  124 


Sumner  navigation,  use  of  bearings 

from  the  bow  in,  134 
Sumner  observations,  motion  of  ship 

between  two,  137 
Sumner  point,  112 

ship  not  at,  125 

Sun,  advance  preparation  for  obser- 
vation of,  92 

altitude  correction  of,  70 

azimuth  of,  44 

compass  bearing  of,  44 

declination  of,  75,  82,  85 

in  nautical  almanac,  76 

its  disk  in  telescope,  67 

lower  limb  of,  71 

mean,  77,  83 

midnight,  observed  for  latitude,  89 

observed  for  index  error,  67 

observed  for  Sumner  line,  112 

right,  ascension  of,  83,  85. 
Superintendent  of  documents,  75 
Superposed  objects  in  sextant,  64 
Supplementary    correction    of    sun's 

altitude,  72 

Swinging  ship  for    compass    adjust- 
ment, 43 

Table,  auxiliary  azimuth,  115 

azimuth,  111 

compass  points,  52 

corrections  for  altitudes,  72 

deviation,  48 

deviation,  second,  49 

dip,  73 

haversines,  99 

logarithms,  25 

logarithms,  inverse  use  of,  28 

mathematical,  10 

mathematical,  inverse  use  of,  12 

multiplication,  11,  28 

of  steering-courses,  51 

Sumner,  by  Kelvin,  126 

Sumner  intersection,  135 

traverse,  10 

traverse,  used  to  convert  longitude 
difference  into  departure,  19 

trigonometric  logarithms,  32 
Tabular,  difference,  12,  27 

numbers,  10 
Taking  a  departure,  53 
Tangency  of  sun's  images,  67,  70 
Tangent,  images  of  sun,  67,  70 

in  trigonometric  tables,  31 

screw  of  sextant,  62 


330 


INPEX 


Telescope,  inverting,  72 

of  sextant,  61 

Ten,  arbitrary  increase  of  logs  by,  25, 
31,  32 

multiples  of,  logs  of,  24 
Time,  apparent  solar,  77 

apparent    solar,    used    for    ship's 
clock,  94 

astronomical,  78 

civil,  78,  93 

difference,  80,  87 

equation  of,  75,  77 

Greenwich  mean,  75 

mean  solar,  77 

sidereal,  85 

solar,  75 

transformation,   mean  to  sidereal, 

85 
Time-sight,  99 

calculation,  latitude  for,  101 

compared  with  Sumner  navigation, 

124 

Transformations,    time,    importance 
of,  82 

mean  to  sidereal,  85 
Transit,  lower,  89 

upper,  89 
Traverse,  sailing,  39 

table,  10 

table,  in  Sumner  method,  113,  135 

table,    used   to    convert   longitude 
difference  and  departure,  19 


Triangle,  parts  of,  31 
Trigonometry,  log  table,  31 

plane,  10,  31 

Tropics,  noon-sight  in,  89 
True  course  of  ship,  42 

north,  43 

Unit  for  measuring  angles,  9 
Upper  transit,  89 
time  of,  for  stars,  96 

Variation  of  compass,  43 

older  designation  of,  47 

when  negative,  47 
Vernal  equinox,  85 
Vernier,  backward  reading,  off  arc,  66 

of  sextant,  64 
Vertex  of  angle  denned,  8 

Watch,  chronometer  minus,  94 

how  set,  94 

used  for  timing  observations,  94 
Whole  number  part  of  log,  23 

rule  for,  24 
Wind,  deflects  ship,  2 

Zenith,  denned,  61 

sun  near  it  in  tropics,  89 
Zero,  hours,  same  as  noon,  75 

meridian,  Greenwich,  3 


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An  Introduction  to  Astronomy 

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MACMILLAN  BOOKS   ON  ASTRONOMY 
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How  to  Identify  the  Stars 

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